High School Mathematics
all, the formula】 sealed, never
Mathematics 【High School, all, the formula】
sin2α = 2sinαcosα
cos2α = cos2α -sin2α = 2cos2α-1 = 1-2sin2α
2tanα
tan2α =-----
1-tan2α
sin3α = 3sinα- 4sin3α
cos3α = 4cos3α-3cosα
3tanα-tan3α
tan3α =------
1-3tan2α
trigonometric functions and the difference of the plot formula of the product of trigonometric functions and differential equations
α + β α-β
sinα + sinβ = 2sin --- · cos --- < br> α + β α-β
sinα-sinβ = 2cos --- · sin ---
α + β α-β
cosα + cosβ = 2cos --- · cos ---
α + β α-β
cosα-cosβ =- 2sin --- · sin ---
sinα · cosβ =- [sin (α + β) + sin (α-β)]
cosα · sinβ =- [sin (α + β)-sin (α-β)]
cosα · cosβ =- [ ,],[angle in the form of a trigonometric function (auxiliary angle trigonometric formulas
collection, a collection of simple logic functions
any x ∈ A x ∈ B, denoted by A B
AB, BAA = B
AB = x
AB = x ∈ A, or x ∈ B ;
card (AB) = card (A) + card (B)-card (AB)
(1) the original proposition if the proposition p is
q
converse, if not q then p
if p then q
proposition against any proposition, if q, then p
(2) the four propositions relationship
(3) AB, A sufficient condition is established B
BA, A is a necessary condition for the establishment of B
AB, A is B established necessary and sufficient condition
Functions Logarithmic
(1) domain, range, corresponding to the rule
(2) monotonicity ;
For any x1, x2 ∈ D
if x1
x1 f (x2), called f (x) in D is a decreasing function
(3) parity
the function f (x) of within the definition of any one of x, if f (-x) = f (x), say f (x) is even function
if f (-x) =- f (x), called f ( x) is an odd function
(4) periodic
for the function f (x) within the definition of any one of x, if there exists a constant T, so f (x + T) = f (x), called f (x) is a periodic function (1) Score Index Score Index of the power
is the meaning of power indices
negative score is the meaning of power ;
(2) of the nature and number of algorithms
loga (MN) = logaM + logaN
logaMn = nlogaM (n ∈ R) < br> exponential function logarithmic function
(1) y = ax (a> 0, a ≠ 1) is called exponential function
(2) x ∈ R, y> 0
image through the (0,1)
a> 1 时, x> 0, y> 1; x <0,0
<1>0 < a <1 时, x> 0,0 <0, x <1;>1
a> 1 时, y = ax is an increasing function of
0 < is a function y="ax" (a (1) decreasing 1,>0, a ≠ 1) is called the logarithmic function
(2) x> 0, y ∈ R
After Image (1,0)
a> 1 时, x> 1, y> 0; 0
<0 <1,>0 <1 x 时,>1, y <0; 0 <1,>0
a> 1 时, y = logax is an increasing function of
0
<1 y="logax" decreasing 时,>exponential equations and logarithmic equations
basic
logaf (x) = bf (x) = ab (a> 0, a ≠ 1)
with the bottom type
logaf (x) = logag (x) f (x) = g (x)> 0 (a> 0, a ≠ 1)
substitution type f (ax) = 0 or f (logax) = 0
series
the basic concepts of arithmetic series, series
(1) Number Sequence formulas an = f (n)
(2) series recurrence formula
(3) series with a general formula of n items and the relationship between < br> an +1- an = d
an = a1 + (n-1) d
a, A, b into the arithmetic 2A = a + b
m + n = k + l am + an = ak + al
commonly used geometric series sum formula
an = a1qn_1
a, G, b into the geometric G2 = ab
m + n = k + l aman = akal
inequality
important inequalities of the basic properties of inequalities
a> bb < a
a> b, b> ca> c
a> b a + c> b + c
a + b> ca> c-b
a> b, c> d a + c> b + d
a> b, c> 0 ac> bc
a> b, c <0 ac
a> b> 0, c> d> 0 ac
a> b> 0 dn> bn (n ∈ Z, n> 1)
a > b> 0> (n ∈ Z, n> 1)
(a-b) 2 ≥ 0
a, b ∈ R a2 + b2 ≥ 2ab
| a | - | b | ≤ | a ± b | ≤ | a | + | b | the basic method of Inequality
comparison
(1) To prove the inequality a> b (or a
a-b> 0 (or a-b <0 = can
(2) If b > 0, to permit a> b, only need to prove that,
to permit a
synthesis method is the synthesis method known or has been proven from the inequality , the realities of the nature of inequality derived want to permit inequality (from the fruit by derivative) method.
analysis method is a sufficient condition for concluding the establishment of start, step by step for the establishment of the necessary and sufficient conditions conditions, until the desired conditions are known correctly so far, apparent in c, b = d
(a + bi) + (c + di) = (a + c) + (b + d) i
(a + bi) - (c + di) = (a-c) + (b-d) i
(a + bi) (c + di) = (ac-bd) + (bc + ad) i
a + bi = r (cosθ + isinθ)
r1 = (cosθ1 + isinθ1)? r2 (cosθ2 + isinθ2)
= r1? r2 〔cos (θ1 + θ2) + isin (θ1 + θ2)
〕 〔r (cosθ + sinθ)〕 n = rn (cosnθ + isinnθ)
k = 0,1, ... ..., n-1 < br> Analytic Geometry
1, line
distance between two points, will score points straight line equation
| AB | = | |
| P1P2 | =
y-y1 = k (x-x1)
y = kx + b
angle between two lines of the location and distance
or k1 = k2, and b1 ≠ b2
l1 and l2 coincidence
or k1 = k2 and b1 = b2
l1 and l2 intersect ;
or k1 ≠ k2
l2 ⊥ l2
or k1k2 =- 1 l1 to l2 corner angle
l1 and l2
point to a line distance
2.
conic
ellipse standard equation of circle (x-a) 2 + (y -b) 2 = r2
center for the (a, b), radius R
general equation x2 + y2 + Dx + Ey + F = 0
where the center is (),
radius r
(1) with a straight-line distance from the center to the radius r d and the judge or the discriminant to determine the location of straight lines and circular relationship
(2) two-round relationship between the position of the radius with the center distance and difference d with the ellipse
determine the focus of F1 (-c, 0), F2 (c, 0)
(b2 = a2-c2)
eccentricity alignment equation
focus radius | MF1 | = a + ex0, | MF2 | = a-ex0
hyperbolic parabola
hyperbolic
focus F1 (-c, 0), F2 (c, 0)
(a, b> 0, b2 = c2-a2 )
eccentricity alignment equation
focus radius | MF1 | = ex0 + a, | MF2 | = ex0-a parabola y2 = 2px (p> 0 )
focus F
alignment axis translation equation
here (h, k) is the origin of the new coordinate system in the original coordinate system of coordinates.
1. a collection of elements each with the opposite sex ① ② ③ deterministic randomness
2. collection representation ① description of enumeration method
③ ② ④ Wayne graph the number of axis method
3. collection operations
⑴ A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
⑵ Cu (A ∩ B) = CuA ∪ CuB
Cu (A ∪ B) = CuA ∩ CuB
4. a collection of the nature of the
⑴ n element subset of the set number: 2n
subset number: 2n-1; non-empty subset number: 2n-2
High School Summary
four mathematical concepts, inequalities
1, if n is a positive odd number, can be introduced by the it? (can)
n is a positive even if it? (non-negative only when)
2, the same can be subtracted to the inequality, divide up (can not)
can add it? (can)
can multiply it? ; (energy, but conditional)
3, two positive numbers mean inequality is:
three positive numbers mean inequality is:
n a positive mean inequality is:
4, the harmonic mean of two positive numbers, geometric mean, arithmetic mean, root mean square is the relationship between the
6, two-way inequality is:
the left of the equal sign in time to obtain the right to obtain equal in time.
five series
1, etc. Arithmetic general formulas, the former n items and formula: =.
2, the general term geometric series formula,
ago n items and the formula is:
3, When the common ratio of geometric series q satisfy <1, = S =. In general, if the infinite series and limits of the former existence of n items, put the limit of the known and the series (or all items and), with S, ie S =.
4, if m, n, p, q ∈ N, and, then: when the series is the arithmetic progression, there; when the series is the geometric sequence, there .
5, arithmetic sequence, if Sn = 10, S2n = 30, then S3n = 60;
6, geometric sequence, if Sn = 10, S2n = 30, then S3n = 70; < br> VI complex
1, how to calculate? (n is 4, except the first request from the remainder,)
2, is one of the two imaginary cube root, and:
3, the complex is set within the triangle inequality:, where the left side in the complex z1, z2 corresponding vector collinear and opposite (in the same direction) is taken equal sign, right in the complex z1, z2, and the corresponding vector collinear with the (reverse) to take equal time.
4, Di Mo Buddha theorem is:
5, if not zero complex number, z n has n-th root, namely:
them in the complex plane the corresponding point in the distribution of any special relationship?
are located in the center of the circle at the origin, radius of the circle and this circle n equal portions.
6, if, complex z1, z2 are corresponding points A, B, then △ AOB (O as the coordinate origin) of the area is.
8, the complex plane z corresponds to the complex a few basic tracks:
① track as a ray. < br> ② path as a ray.
③ trajectory is a circle.
④ trajectory is a straight line.
⑤ There are three possible trajectories circumstances: a) At the time, oval track; b) At the time, track is a line segment; c) At the time, path does not exist.
⑥ track there are three possible cases: a) time, the trajectory is hyperbolic; b) At the time, for the two-ray path; c) At the time, path does not exist.
seven, permutations and combinations, binomial theorem
1, addition principle, multiplication principle applies to what the situation? What are the characteristics?
additive category, class, class of independence; multiplication step, step by step related.
2, Permutation formula is: = =;
arrangement number and the relationship between the number of combinations:
combinations formula is: = =;
; combination of several properties: = + =
3, the binomial theorem: expansion of two general formulas:
eight, analytic geometry
1, Shull formula:
2, number line distance between two points formula:
3, rectangular coordinate plane distance formula between two points:
4, if the point P divided into a fixed ratio to the line λ, then λ =
5, if the points, point P is divided into a fixed ratio to the line λ, is: λ = =;
If, then △ ABC the coordinates of the center of G is.
6, Line Slope was defined as k =, two formula k =.
7, several forms of linear equations:
point slope form:, inclined cutting:
two-point: the intercept-type:
; general formula:
the intersection of two lines through the linear system equation is:
14, of circular and linear relationship between the location of the most commonly used in two ways , that is:
① discriminant method: Δ> 0, = 0, <0, equivalent to a straight line and circle intersect, tangent, Deviation;
; ② examine the center of the circle to the line between distance and size of the radius: the distance is greater than the radius, equal to the radius, smaller than the radius, equivalent to phase away from the straight line and circle, tangent, intersect.
15, the four parabolic form of the standard equation:
16, the focus of the parabolic coordinates are:, quasi-line equation is:.
If the point is a point on the parabola, the focal point of the distance to the parabola (called the focal radius) is:, over the focus of the parabola and the parabola symmetry axis perpendicular to the string ( known as Path) in length are:.
17, ellipse standard equation of the two forms are: and
.
18, the focus of elliptical coordinates, the quasi-line equation is that eccentricity is the long path is. Them.
19, if the point is elliptic on the point is that its left and right focus, the focal point P and the radius length is.
20, the standard hyperbolic equations in two forms: and
.
21, the focus of the hyperbolic coordinates, the quasi-line equation is that eccentricity is a long path, the asymptotic line equation is. Them.
22, with a total of asymptote hyperbola hyperbolic equation system is. And hyperbolic equations were the focus of the hyperbolic system is.
23, if the straight lines and conic curves intersect at two points A (x1, y1), B (x2, y2), then the string length;
If the straight lines and conic curves intersect at two points A (x1, y1), B (x2, y2), then the string length.
24, the focal conic geometric parameter p is the significance to the reference line of the focus distance, the ellipse and hyperbola are:.
25, translation axis, the origin of the new coordinate system in the original coordinate system coordinates (h, k), if the point P in the original coordinate system the coordinates of the coordinate system in the new coordinates is then = =.
second, composite second radical simplification of
When is a completely square, on the shape of the radical reduction using the above formulation is more convenient.
⑵ and set the number of elements:
n (A ∪ B) = nA + nB-n (A ∩ B)
5. N natural numbers or non-negative integers
Z set of integers Q R rational real numbers
6. Simple logic truth table
p consistent with the proposition true or false
non-p
false true
in the definition domain, if, for the dual function; if it was odd function.
1 had two points and only a straight line the shortest line between two points
2
3 of the supplementary angle with the same angle or isometric angle or isometric
4 with the complementary angle equal to
5 had a little one and only one straight line and known point outside a straight line perpendicular to
6 points connected with straight line segments in all, the vertical section of the shortest
7 point outside a straight line through the parallel axiom, and only one straight line and this line if the two lines parallel to
8 and the third line are parallel, these two lines are parallel to each other
9 Tong Weijiao equal, the two straight lines parallel to the wrong
10 equal angles, two straight lines Parallel
11 complementary with the adjacent angles, two straight lines parallel to the two parallel
12, Tong Weijiao equal
13 two parallel lines, alternate angles are equal within the two lines parallel
14, with the next complementary angles
15 on both sides of the triangle theorems and inference than the third side
16 on both sides of the triangle is less than the third side
17 angles of a triangle and the theorem of the three angles of a triangle and is equal to 180 °
18 Corollary 1 right angle more than two acute triangles each triangle
19 Corollary 2 and it is not an exterior angle is equal to two adjacent interior angles
20 Corollary 3 the triangle exterior angle is greater than any one and it is not adjacent angles
21 corresponding sides congruent triangles, corresponding angles are equal
22 corner edge axiom (SAS) has two sides and their corresponding angles equal two triangles congruent
23 angle corners axiom (ASA ) he had two horns and their corresponding folders sides congruent two triangles of equal
24 Corollary (AAS) which had two horns, and the corresponding corner on the edge of two triangles of equal side of justice congruent
25 Collage (SSS) with the corresponding three sides of equal triangles congruent
26 hypotenuse of two right-angled edge of axioms (HL) and a right-angle bevel edge with the corresponding two triangles congruent equivalent of Theorem 1 in the corner
27 point to divide equally the angle of the line equidistant from both sides of Theorem 2-1
28 angle from both sides of the same points in the angle bisector of angle bisector
29 is the angle equidistant points on both sides of the set of all the nature
30 isosceles triangle isosceles triangle theorem of the two bottom corners are equal (that is, on the other side isometric)
31 Corollary 1 lines equally divide equally angle isosceles triangle bottom and perpendicular to the bottom of the
32 isosceles triangle the angle bisector, the bottom edge of the middle and the bottom edge of the high overlap each other
33 Corollary 3 corners of an equilateral triangle are equal, and each a corner of the isosceles triangle is equal to 60 °
34 Judgement Theorem If a triangle has two angles equal, then the two angles are equal the right side (equiangular, equilateral)
35 Corollary 1, three corners are equal is equilateral triangle
36 Corollary 2 has an angle equal to 60 ° isosceles triangles are equilateral triangles
37 in a right triangle, if one acute angle is 30 ° so it right right-angle bevel edge is equal to half the hypotenuse of a right triangle
38 the center line on the hypotenuse is equal to half the
39 Theorem on the axis line segment point and the two ends of this segment are equidistant
40 inverse and a line equidistant from the two end points in this segment of the vertical split line
41 perpendicular bisector of line segment and the segment end points can be regarded as equal to all the points from the set of
42 Theorem 1 in a straight line on the symmetry of the two figures are congruent shape
43 Theorem 2 If two graphics symmetrical about a straight line, then the corresponding point of the axis of symmetry is the axis connecting Theorem 3 Two
44 symmetrical about a straight line graph if their corresponding intersection line or extension cord, then the intersection of the inverse of the axis of symmetry
45 points if the two corresponding connection graph is a straight line with the axis, then the two graphics on the this line of two symmetrical
46 Pythagorean triangle edge at right angles a, b and the square is equal to the square of the hypotenuse c, ie a ^ 2 + b ^ 2 = c ^ 2
47 Pythagorean Theorem If the triangle triangular inverse long a, b, c has a relationship a ^ 2 + b ^ 2 = c ^ 2, then the triangle is right triangle quadrilateral theorems
48 equal angles and 360 °
49 quadrilateral exterior angle equal to 360 °
50 angles and the theorem of n sides polygon-shaped interior angles is equal to (n-2) × 180 °
51 deduction equal to any multilateral exterior angle and 360 °
52 parallelogram properties Theorem 1 parallelogram equal
53 diagonal parallelogram parallelogram nature of Theorem 2 on the opposite side in the same folder
54 inference between two parallel lines, parallel line segments are equal
55 parallelogram parallelogram nature of Theorem 3 diagonal to each other equally
56 Theorem 1 determine the two parallelogram angles are equal to the quadrilateral is a parallelogram parallelogram
57 Theorem 2 to determine the respective groups on the same side of a quadrilateral is a parallelogram
58 Theorem 3 diagonal parallelogram to determine each split quadrilateral is a parallelogram parallelogram
59 Theorem 4 to determine a set of parallel sides equal quadrilateral is a parallelogram
60 rectangular nature of Theorem 1, the four corners of the rectangle are rectangular Cartesian
61 rectangular diagonal nature of Theorem 2 is equal
62 has three rectangular determine the angle of Theorem 1 is at right angles to rectangular quadrilateral is a rectangle
63 Theorem 2 to determine the diagonal of the parallelogram is a rectangle equal to < br> 64 diamond Diamond Theorem 1 the nature of the four sides are equal
65 diamond diamond diagonal nature of Theorem 2, perpendicular to each other, and each diagonal split a diamond-shaped area on the corner
66 = diagonal half of the product, ie S = (a × b) ÷ 2
67 diamond-shaped sides are equal Theorem 1 determine a quadrilateral is a rhombus diamond
68 Theorem 2 to determine the diagonal of the parallelogram are perpendicular to each other diamond
Theorem 1 69 square nature of the four corners of a square are right angles, four sides are equal
70 square nature of Theorem 2, the two diagonal squares are equal, and perpendicular to each other equally, and each diagonal a diagonal split
71 Theorem 1 on the center of symmetry of the two graphs are congruent
72 Theorem 2, the two graphics on the center of symmetry, symmetry point connection have been the center of symmetry and is symmetric against the center divide
73 Theorem If two graphs have been a corresponding point of the connection point, and is it equally, then the two graphics on this point the nature of symmetry
74 isosceles trapezoid isosceles trapezoid theorems in the same two corners on the bottom isosceles trapezoid are equal
75 two diagonals are equal decision theorem
76 isosceles trapezoid in the same two corners on the bottom of the ladder are equal isosceles trapezoid
77 diagonal is equal to the trapezoidal isosceles trapezoid
78 equal portions parallel segments Theorem If a set of parallel lines cut in a straight line segment obtained
equal, then the other straight line intercepted the line are equal
79 Corollary 1 through trapezoidal the midpoint of the waist and bottom of a parallel line, the other will be shared equally by the waist
80 Corollary 2 side of the midpoint of the triangle parallel with the other side of the line, will split the third side
81 triangle triangle median line of Theorem the median line parallel to the third side and equal to its half of the
82 trapezoid trapezoid theorem of the median line median line parallel to the two end and two equal to half of the bottom and L = (a + b) ÷ 2 S = L × h
83 (1) If the proportion of the basic properties of a: b = c: d, then ad = bc
if ad = bc, then a: b = c: d wc / S ?
84 (2) the nature of cooperation than if a / b = c / d, then (a ± b) / b = (c ± d) / d
85 (3) If the geometric properties of a / b = c / d = ... = m / n (b + d + ... + n ≠ 0), then
(a + c + ... + m) / (b + d + ... + n) = a / b
86 parallel line segments proportional to the theorem of three sub-parallel lines cut two lines from the corresponding line segments proportional reasoning
87 side of the line parallel to the triangle cut other side (or both sides of the extension cord), which is proportional to the corresponding segment < br> 88 Theorem If a line cut on both sides of the triangle (or both sides of the extension cord) from the corresponding line segments proportional, then this line parallel to the triangle's third side
89 parallel to the side of the triangle, and the other side intersecting lines, triangles intercepted by the triangular sides of a triangle corresponds to the original proportional
90 Theorem side of the line parallel to the triangle and the other side (or both sides of the extension line) intersects the triangle composed of the original triangle Similar
91 corners of similar triangles to determine the corresponding equivalent of Theorem 1, the two triangles similar to the (ASA)
92 high on the hypotenuse of a right triangle is divided into two right triangles and similar triangles of the original decision theorem 2
93 proportional to the corresponding sides and angles equal, the two triangles similar to the (SAS)
94 to determine the corresponding proportion trilateral Theorem 3, the two triangles similar to the (SSS)
95 Theorem If the hypotenuse of a right triangle and a right-angle edge hypotenuse of another right-angle side and a corresponding proportion, then the two triangles similar to the nature of Theorem 1
96 higher than the corresponding similar triangles, corresponding to the center line than the corresponding angle bisector is equal to the ratio nature than
97 similar similar triangles theorem 2 ratio is equal to the circumference of the nature similar to Theorem 3 than
98 similar similar triangles is equal to the area than the square
99 than the sine of any acute angle equal to its complementary angle of the cosine, any acute angle equal to its complementary angle cosine sine
100 tangent of any acute angle equal to its complementary angle cotangent value, the value of any acute angle equal to its complementary angle cotangent tangent of < br> 101 round is a distance equal to the fixed-length fixed-point set of points inside the circle
102 can be seen as the distance is less than the radius of the center point of the set
103 can be seen as outside the circle from the center of the circle greater than the radius of the set of points such
104 circle with the radius of a circle or equal
105 a distance equal to the fixed point of the trajectory of fixed length, is the point of a circle, a circle of radius length
106 and the distance between the two endpoints are known to line the locus of points equal to, is the perpendicular bisector of line segments to a known angle
107 equidistant points on both sides of the track, is the angle bisector
108-2 equidistant parallel lines locus of points, and it is these two parallel lines parallel and equidistant Theorem is not a straight line
109 three-point line has been identified with a circle.
110 vertical diameter perpendicular to the chord diameter of Theorem equally split this string by string and the two arcs
111 of Corollary 1 ① split string (not diameter) perpendicular to the string diameter, and equally the right chord arc
② two perpendicular bisector of string through the center of the circle, and the strings are split on the two strings of the arc
③ split an arc of diameter, the vertical split string, and split the string by another pair Corollary 2 arc
112 round clip of two parallel strings are equal
113 circle arc is the center of the center as the center of symmetry theorem of symmetry
114 or so in the same round in the circle, central angle equal The equivalent of the arc, the equivalent of the string, the string on the heart strings of inferences from the same
115 round or so in the same round, if the two central angle, two arcs, two string or two strings heart strings in a set amount of distance equal to that they correspond to the amount of the other groups are equal
116 theorem of an arc on the circumference of the angle of the central angle is equal to its half
117 Corollary 1 with the arc or other arc of the circumference of the equal angles; the same circle or other circular, the equivalent of the circumferential angle of the arc are equal
118 Corollary 2 semicircle (or diameter) of the angle of the circle at right angles; 90 ° circumferential angle The diameter of the string is
119 Corollary 3 If the center line on the side of the triangle is equal to half the side, then the triangle is right triangle within the circle theorems
120 quadrilateral diagonal complementary, and any exterior angle is is equal to its angle within
121 ① ⊙ O line L and intersects d
② ⊙ O line L, and d = r
③ tangent line L and ⊙ O phase from the d> r?
After 122 Tangent Theorem to determine the radius of the outer end and perpendicular to the radius of this circle tangent line is tangent to the nature of the Theorem
123 round after a cut perpendicular to the tangent point of radius
124 1 through the center of the circle and the inference will be perpendicular to the tangent line through the tangent point
125 Corollary 2 points after a cut perpendicular to the tangent line through the center
126 will be a long tangent point theorems introduced from outside the circle of the two tangent circles, their appearance and so the tangent , center, and split the connection that the angle between two tangents
127 quadrilateral circumscribed circle on the side of the two groups and equal
128 Xian Qiejiao Theorem Xian Qiejiao arc is equal to its folder circumferential angle
129 inference that if two Xianqie Jiao folder arc equal, then the two are equal
130 Xian Qiejiao intersection of the two strings intersect circle theorem of string, was divided into two intersection equal to the product of a long line
131 deduction if the chord and the diameter of vertical intersection, then the string by half its diameter into two sub-segments in the proportion of items
132 cutting line from the circle theorems point outside the circle tangent cited and the secant, tangent length is the secant and the circle to the intersection point of two line segments in the proportion of long-term
133 inferences that lead from outside the circle round the two secant, which is the secant and the circle of each the intersection of the product of two equal length segments if the two circles tangent
134, then the cutoff point must be in line with the heart outside the two circles
135 ① from the d> R + r ② d = exterior of two circles R + r
③ two circles intersect Rr r)
④ two circles inscribed d = Rr (R> r) ⑤ two circles containing d r)
136 two-circle theorem of intersecting lines with the axis of two circles the heart of public * String
137 into the circle theorem of n (n ≥ 3):
⑴ points followed by links from each polygon is The circle inscribed n-gon
⑵ points for each round after the tangent to the intersection of the tangent to the adjacent vertices of this polygon is a circle circumscribed n-gon theorem of any regular polygon
138 has a circumcircle and an inscribed circle, the two circles are concentric
139 regular n-gon of each interior angle is equal to (n-2) × 180 ° / n
140 theorems are n edge Heart-shaped edge radius and n from the regular 2n-gon into two congruent right triangle
141 n-gon is the area of Sn = pnrn / 2 p n-gon that is the perimeter of an equilateral triangle
142 area of √ 3a / 4 a side said
143 around if a vertex k-regular n-gon angle, and these angles should be 360 °, so k × (n-2) 180 ° / n = 360 ° into the (n-2) (k-2) = 4
144 arc length formula: L = n Wu R/180
145 fan-shaped area formula: S fan = n Wu R ^ 2 / 360 = LR / 2
146 within the common tangent length = d-(Rr) grandfather tangent length = d-(R + r)
multiplication and factoring
a ^ 2-b ^ 2 = ( a + b) (ab)
a ^ 3 + b ^ 3 = (a + b) (a ^ 2-ab + b ^ 2)
a ^ 3-b ^ 3 = (ab (a ^ 2 + ab + b ^ 2)
triangle inequality | a + b | ≤ | a | + | b | | ab | ≤ | a | + | b | | a | ≤ b <=>-b ≤ a ≤ b
| ab | ≥ | a | - | b | - | a | ≤ a ≤ | a |
the solution of a quadratic equation-b + √ (b ^ 2-4ac) / 2a-b- √ (b ^ 2-4ac) / 2a
relationship between roots and coefficients X1 + X2 =- b / a X1 * X2 = c / a Note: Whyte Theorem
discriminant
b ^ 2 - 4ac = 0 Note: The equation has two equal real roots
b ^ 2-4ac> 0 Note: The equation has two unequal real roots
b ^ 2-4ac <0 Note: The equation has no real root, root
a conjugate trigonometric formulas
corners and formulas
sin (A + B) = sinAcosB + cosAsinB
sin (AB) = sinAcosB-sinBcosA
cos ( A + B) = cosAcosB-sinAsinB
cos (AB) = cosAcosB + sinAsinB
tan (A + B) = (tanA + tanB) / (1-tanAtanB)
tan (AB) = (tanA -tanB) / (1 + tanAtanB)
cot (A + B) = (cotAcotB-1) / (cotB + cotA)
cot (AB) = (cotAcotB +1) / (cotB-cotA).
all, the formula】 sealed, never
Mathematics 【High School, all, the formula】
sin2α = 2sinαcosα
cos2α = cos2α -sin2α = 2cos2α-1 = 1-2sin2α
2tanα
tan2α =-----
1-tan2α
sin3α = 3sinα- 4sin3α
cos3α = 4cos3α-3cosα
3tanα-tan3α
tan3α =------
1-3tan2α
trigonometric functions and the difference of the plot formula of the product of trigonometric functions and differential equations
α + β α-β
sinα + sinβ = 2sin --- · cos --- < br> α + β α-β
sinα-sinβ = 2cos --- · sin ---
α + β α-β
cosα + cosβ = 2cos --- · cos ---
α + β α-β
cosα-cosβ =- 2sin --- · sin ---
sinα · cosβ =- [sin (α + β) + sin (α-β)]
cosα · sinβ =- [sin (α + β)-sin (α-β)]
cosα · cosβ =- [ ,],[angle in the form of a trigonometric function (auxiliary angle trigonometric formulas
collection, a collection of simple logic functions
any x ∈ A x ∈ B, denoted by A B
AB, BAA = B
AB = x
AB = x ∈ A, or x ∈ B ;
card (AB) = card (A) + card (B)-card (AB)
(1) the original proposition if the proposition p is
q
converse, if not q then p
if p then q
proposition against any proposition, if q, then p
(2) the four propositions relationship
(3) AB, A sufficient condition is established B
BA, A is a necessary condition for the establishment of B
AB, A is B established necessary and sufficient condition
Functions Logarithmic
(1) domain, range, corresponding to the rule
(2) monotonicity ;
For any x1, x2 ∈ D
if x1
x1 f (x2), called f (x) in D is a decreasing function
(3) parity
the function f (x) of within the definition of any one of x, if f (-x) = f (x), say f (x) is even function
if f (-x) =- f (x), called f ( x) is an odd function
(4) periodic
for the function f (x) within the definition of any one of x, if there exists a constant T, so f (x + T) = f (x), called f (x) is a periodic function (1) Score Index Score Index of the power
is the meaning of power indices
negative score is the meaning of power ;
(2) of the nature and number of algorithms
loga (MN) = logaM + logaN
logaMn = nlogaM (n ∈ R) < br> exponential function logarithmic function
(1) y = ax (a> 0, a ≠ 1) is called exponential function
(2) x ∈ R, y> 0
image through the (0,1)
a> 1 时, x> 0, y> 1; x <0,0
<1>0 < a <1 时, x> 0,0 <0, x <1;>1
a> 1 时, y = ax is an increasing function of
0 < is a function y="ax" (a (1) decreasing 1,>0, a ≠ 1) is called the logarithmic function
(2) x> 0, y ∈ R
After Image (1,0)
a> 1 时, x> 1, y> 0; 0
<0 <1,>0 <1 x 时,>1, y <0; 0 <1,>0
a> 1 时, y = logax is an increasing function of
0
<1 y="logax" decreasing 时,>exponential equations and logarithmic equations
basic
logaf (x) = bf (x) = ab (a> 0, a ≠ 1)
with the bottom type
logaf (x) = logag (x) f (x) = g (x)> 0 (a> 0, a ≠ 1)
substitution type f (ax) = 0 or f (logax) = 0
series
the basic concepts of arithmetic series, series
(1) Number Sequence formulas an = f (n)
(2) series recurrence formula
(3) series with a general formula of n items and the relationship between < br> an +1- an = d
an = a1 + (n-1) d
a, A, b into the arithmetic 2A = a + b
m + n = k + l am + an = ak + al
commonly used geometric series sum formula
an = a1qn_1
a, G, b into the geometric G2 = ab
m + n = k + l aman = akal
inequality
important inequalities of the basic properties of inequalities
a> bb < a
a> b, b> ca> c
a> b a + c> b + c
a + b> ca> c-b
a> b, c> d a + c> b + d
a> b, c> 0 ac> bc
a> b, c <0 ac
a> b> 0, c> d> 0 ac
a> b> 0 dn> bn (n ∈ Z, n> 1)
a > b> 0> (n ∈ Z, n> 1)
(a-b) 2 ≥ 0
a, b ∈ R a2 + b2 ≥ 2ab
| a | - | b | ≤ | a ± b | ≤ | a | + | b | the basic method of Inequality
comparison
(1) To prove the inequality a> b (or a
a-b> 0 (or a-b <0 = can
(2) If b > 0, to permit a> b, only need to prove that,
to permit a
synthesis method is the synthesis method known or has been proven from the inequality , the realities of the nature of inequality derived want to permit inequality (from the fruit by derivative) method.
analysis method is a sufficient condition for concluding the establishment of start, step by step for the establishment of the necessary and sufficient conditions conditions, until the desired conditions are known correctly so far, apparent in c, b = d
(a + bi) + (c + di) = (a + c) + (b + d) i
(a + bi) - (c + di) = (a-c) + (b-d) i
(a + bi) (c + di) = (ac-bd) + (bc + ad) i
a + bi = r (cosθ + isinθ)
r1 = (cosθ1 + isinθ1)? r2 (cosθ2 + isinθ2)
= r1? r2 〔cos (θ1 + θ2) + isin (θ1 + θ2)
〕 〔r (cosθ + sinθ)〕 n = rn (cosnθ + isinnθ)
k = 0,1, ... ..., n-1 < br> Analytic Geometry
1, line
distance between two points, will score points straight line equation
| AB | = | |
| P1P2 | =
y-y1 = k (x-x1)
y = kx + b
angle between two lines of the location and distance
or k1 = k2, and b1 ≠ b2
l1 and l2 coincidence
or k1 = k2 and b1 = b2
l1 and l2 intersect ;
or k1 ≠ k2
l2 ⊥ l2
or k1k2 =- 1 l1 to l2 corner angle
l1 and l2
point to a line distance
2.
conic
ellipse standard equation of circle (x-a) 2 + (y -b) 2 = r2
center for the (a, b), radius R
general equation x2 + y2 + Dx + Ey + F = 0
where the center is (),
radius r
(1) with a straight-line distance from the center to the radius r d and the judge or the discriminant to determine the location of straight lines and circular relationship
(2) two-round relationship between the position of the radius with the center distance and difference d with the ellipse
determine the focus of F1 (-c, 0), F2 (c, 0)
(b2 = a2-c2)
eccentricity alignment equation
focus radius | MF1 | = a + ex0, | MF2 | = a-ex0
hyperbolic parabola
hyperbolic
focus F1 (-c, 0), F2 (c, 0)
(a, b> 0, b2 = c2-a2 )
eccentricity alignment equation
focus radius | MF1 | = ex0 + a, | MF2 | = ex0-a parabola y2 = 2px (p> 0 )
focus F
alignment axis translation equation
here (h, k) is the origin of the new coordinate system in the original coordinate system of coordinates.
1. a collection of elements each with the opposite sex ① ② ③ deterministic randomness
2. collection representation ① description of enumeration method
③ ② ④ Wayne graph the number of axis method
3. collection operations
⑴ A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
⑵ Cu (A ∩ B) = CuA ∪ CuB
Cu (A ∪ B) = CuA ∩ CuB
4. a collection of the nature of the
⑴ n element subset of the set number: 2n
subset number: 2n-1; non-empty subset number: 2n-2
High School Summary
four mathematical concepts, inequalities
1, if n is a positive odd number, can be introduced by the it? (can)
n is a positive even if it? (non-negative only when)
2, the same can be subtracted to the inequality, divide up (can not)
can add it? (can)
can multiply it? ; (energy, but conditional)
3, two positive numbers mean inequality is:
three positive numbers mean inequality is:
n a positive mean inequality is:
4, the harmonic mean of two positive numbers, geometric mean, arithmetic mean, root mean square is the relationship between the
6, two-way inequality is:
the left of the equal sign in time to obtain the right to obtain equal in time.
five series
1, etc. Arithmetic general formulas, the former n items and formula: =.
2, the general term geometric series formula,
ago n items and the formula is:
3, When the common ratio of geometric series q satisfy <1, = S =. In general, if the infinite series and limits of the former existence of n items, put the limit of the known and the series (or all items and), with S, ie S =.
4, if m, n, p, q ∈ N, and, then: when the series is the arithmetic progression, there; when the series is the geometric sequence, there .
5, arithmetic sequence, if Sn = 10, S2n = 30, then S3n = 60;
6, geometric sequence, if Sn = 10, S2n = 30, then S3n = 70; < br> VI complex
1, how to calculate? (n is 4, except the first request from the remainder,)
2, is one of the two imaginary cube root, and:
3, the complex is set within the triangle inequality:, where the left side in the complex z1, z2 corresponding vector collinear and opposite (in the same direction) is taken equal sign, right in the complex z1, z2, and the corresponding vector collinear with the (reverse) to take equal time.
4, Di Mo Buddha theorem is:
5, if not zero complex number, z n has n-th root, namely:
them in the complex plane the corresponding point in the distribution of any special relationship?
are located in the center of the circle at the origin, radius of the circle and this circle n equal portions.
6, if, complex z1, z2 are corresponding points A, B, then △ AOB (O as the coordinate origin) of the area is.
8, the complex plane z corresponds to the complex a few basic tracks:
① track as a ray. < br> ② path as a ray.
③ trajectory is a circle.
④ trajectory is a straight line.
⑤ There are three possible trajectories circumstances: a) At the time, oval track; b) At the time, track is a line segment; c) At the time, path does not exist.
⑥ track there are three possible cases: a) time, the trajectory is hyperbolic; b) At the time, for the two-ray path; c) At the time, path does not exist.
seven, permutations and combinations, binomial theorem
1, addition principle, multiplication principle applies to what the situation? What are the characteristics?
additive category, class, class of independence; multiplication step, step by step related.
2, Permutation formula is: = =;
arrangement number and the relationship between the number of combinations:
combinations formula is: = =;
; combination of several properties: = + =
3, the binomial theorem: expansion of two general formulas:
eight, analytic geometry
1, Shull formula:
2, number line distance between two points formula:
3, rectangular coordinate plane distance formula between two points:
4, if the point P divided into a fixed ratio to the line λ, then λ =
5, if the points, point P is divided into a fixed ratio to the line λ, is: λ = =;
If, then △ ABC the coordinates of the center of G is.
6, Line Slope was defined as k =, two formula k =.
7, several forms of linear equations:
point slope form:, inclined cutting:
two-point: the intercept-type:
; general formula:
the intersection of two lines through the linear system equation is:
14, of circular and linear relationship between the location of the most commonly used in two ways , that is:
① discriminant method: Δ> 0, = 0, <0, equivalent to a straight line and circle intersect, tangent, Deviation;
; ② examine the center of the circle to the line between distance and size of the radius: the distance is greater than the radius, equal to the radius, smaller than the radius, equivalent to phase away from the straight line and circle, tangent, intersect.
15, the four parabolic form of the standard equation:
16, the focus of the parabolic coordinates are:, quasi-line equation is:.
If the point is a point on the parabola, the focal point of the distance to the parabola (called the focal radius) is:, over the focus of the parabola and the parabola symmetry axis perpendicular to the string ( known as Path) in length are:.
17, ellipse standard equation of the two forms are: and
.
18, the focus of elliptical coordinates, the quasi-line equation is that eccentricity is the long path is. Them.
19, if the point is elliptic on the point is that its left and right focus, the focal point P and the radius length is.
20, the standard hyperbolic equations in two forms: and
.
21, the focus of the hyperbolic coordinates, the quasi-line equation is that eccentricity is a long path, the asymptotic line equation is. Them.
22, with a total of asymptote hyperbola hyperbolic equation system is. And hyperbolic equations were the focus of the hyperbolic system is.
23, if the straight lines and conic curves intersect at two points A (x1, y1), B (x2, y2), then the string length;
If the straight lines and conic curves intersect at two points A (x1, y1), B (x2, y2), then the string length.
24, the focal conic geometric parameter p is the significance to the reference line of the focus distance, the ellipse and hyperbola are:.
25, translation axis, the origin of the new coordinate system in the original coordinate system coordinates (h, k), if the point P in the original coordinate system the coordinates of the coordinate system in the new coordinates is then = =.
second, composite second radical simplification of
When is a completely square, on the shape of the radical reduction using the above formulation is more convenient.
⑵ and set the number of elements:
n (A ∪ B) = nA + nB-n (A ∩ B)
5. N natural numbers or non-negative integers
Z set of integers Q R rational real numbers
6. Simple logic truth table
p consistent with the proposition true or false
non-p
false true
in the definition domain, if, for the dual function; if it was odd function.
1 had two points and only a straight line the shortest line between two points
2
3 of the supplementary angle with the same angle or isometric angle or isometric
4 with the complementary angle equal to
5 had a little one and only one straight line and known point outside a straight line perpendicular to
6 points connected with straight line segments in all, the vertical section of the shortest
7 point outside a straight line through the parallel axiom, and only one straight line and this line if the two lines parallel to
8 and the third line are parallel, these two lines are parallel to each other
9 Tong Weijiao equal, the two straight lines parallel to the wrong
10 equal angles, two straight lines Parallel
11 complementary with the adjacent angles, two straight lines parallel to the two parallel
12, Tong Weijiao equal
13 two parallel lines, alternate angles are equal within the two lines parallel
14, with the next complementary angles
15 on both sides of the triangle theorems and inference than the third side
16 on both sides of the triangle is less than the third side
17 angles of a triangle and the theorem of the three angles of a triangle and is equal to 180 °
18 Corollary 1 right angle more than two acute triangles each triangle
19 Corollary 2 and it is not an exterior angle is equal to two adjacent interior angles
20 Corollary 3 the triangle exterior angle is greater than any one and it is not adjacent angles
21 corresponding sides congruent triangles, corresponding angles are equal
22 corner edge axiom (SAS) has two sides and their corresponding angles equal two triangles congruent
23 angle corners axiom (ASA ) he had two horns and their corresponding folders sides congruent two triangles of equal
24 Corollary (AAS) which had two horns, and the corresponding corner on the edge of two triangles of equal side of justice congruent
25 Collage (SSS) with the corresponding three sides of equal triangles congruent
26 hypotenuse of two right-angled edge of axioms (HL) and a right-angle bevel edge with the corresponding two triangles congruent equivalent of Theorem 1 in the corner
27 point to divide equally the angle of the line equidistant from both sides of Theorem 2-1
28 angle from both sides of the same points in the angle bisector of angle bisector
29 is the angle equidistant points on both sides of the set of all the nature
30 isosceles triangle isosceles triangle theorem of the two bottom corners are equal (that is, on the other side isometric)
31 Corollary 1 lines equally divide equally angle isosceles triangle bottom and perpendicular to the bottom of the
32 isosceles triangle the angle bisector, the bottom edge of the middle and the bottom edge of the high overlap each other
33 Corollary 3 corners of an equilateral triangle are equal, and each a corner of the isosceles triangle is equal to 60 °
34 Judgement Theorem If a triangle has two angles equal, then the two angles are equal the right side (equiangular, equilateral)
35 Corollary 1, three corners are equal is equilateral triangle
36 Corollary 2 has an angle equal to 60 ° isosceles triangles are equilateral triangles
37 in a right triangle, if one acute angle is 30 ° so it right right-angle bevel edge is equal to half the hypotenuse of a right triangle
38 the center line on the hypotenuse is equal to half the
39 Theorem on the axis line segment point and the two ends of this segment are equidistant
40 inverse and a line equidistant from the two end points in this segment of the vertical split line
41 perpendicular bisector of line segment and the segment end points can be regarded as equal to all the points from the set of
42 Theorem 1 in a straight line on the symmetry of the two figures are congruent shape
43 Theorem 2 If two graphics symmetrical about a straight line, then the corresponding point of the axis of symmetry is the axis connecting Theorem 3 Two
44 symmetrical about a straight line graph if their corresponding intersection line or extension cord, then the intersection of the inverse of the axis of symmetry
45 points if the two corresponding connection graph is a straight line with the axis, then the two graphics on the this line of two symmetrical
46 Pythagorean triangle edge at right angles a, b and the square is equal to the square of the hypotenuse c, ie a ^ 2 + b ^ 2 = c ^ 2
47 Pythagorean Theorem If the triangle triangular inverse long a, b, c has a relationship a ^ 2 + b ^ 2 = c ^ 2, then the triangle is right triangle quadrilateral theorems
48 equal angles and 360 °
49 quadrilateral exterior angle equal to 360 °
50 angles and the theorem of n sides polygon-shaped interior angles is equal to (n-2) × 180 °
51 deduction equal to any multilateral exterior angle and 360 °
52 parallelogram properties Theorem 1 parallelogram equal
53 diagonal parallelogram parallelogram nature of Theorem 2 on the opposite side in the same folder
54 inference between two parallel lines, parallel line segments are equal
55 parallelogram parallelogram nature of Theorem 3 diagonal to each other equally
56 Theorem 1 determine the two parallelogram angles are equal to the quadrilateral is a parallelogram parallelogram
57 Theorem 2 to determine the respective groups on the same side of a quadrilateral is a parallelogram
58 Theorem 3 diagonal parallelogram to determine each split quadrilateral is a parallelogram parallelogram
59 Theorem 4 to determine a set of parallel sides equal quadrilateral is a parallelogram
60 rectangular nature of Theorem 1, the four corners of the rectangle are rectangular Cartesian
61 rectangular diagonal nature of Theorem 2 is equal
62 has three rectangular determine the angle of Theorem 1 is at right angles to rectangular quadrilateral is a rectangle
63 Theorem 2 to determine the diagonal of the parallelogram is a rectangle equal to < br> 64 diamond Diamond Theorem 1 the nature of the four sides are equal
65 diamond diamond diagonal nature of Theorem 2, perpendicular to each other, and each diagonal split a diamond-shaped area on the corner
66 = diagonal half of the product, ie S = (a × b) ÷ 2
67 diamond-shaped sides are equal Theorem 1 determine a quadrilateral is a rhombus diamond
68 Theorem 2 to determine the diagonal of the parallelogram are perpendicular to each other diamond
Theorem 1 69 square nature of the four corners of a square are right angles, four sides are equal
70 square nature of Theorem 2, the two diagonal squares are equal, and perpendicular to each other equally, and each diagonal a diagonal split
71 Theorem 1 on the center of symmetry of the two graphs are congruent
72 Theorem 2, the two graphics on the center of symmetry, symmetry point connection have been the center of symmetry and is symmetric against the center divide
73 Theorem If two graphs have been a corresponding point of the connection point, and is it equally, then the two graphics on this point the nature of symmetry
74 isosceles trapezoid isosceles trapezoid theorems in the same two corners on the bottom isosceles trapezoid are equal
75 two diagonals are equal decision theorem
76 isosceles trapezoid in the same two corners on the bottom of the ladder are equal isosceles trapezoid
77 diagonal is equal to the trapezoidal isosceles trapezoid
78 equal portions parallel segments Theorem If a set of parallel lines cut in a straight line segment obtained
equal, then the other straight line intercepted the line are equal
79 Corollary 1 through trapezoidal the midpoint of the waist and bottom of a parallel line, the other will be shared equally by the waist
80 Corollary 2 side of the midpoint of the triangle parallel with the other side of the line, will split the third side
81 triangle triangle median line of Theorem the median line parallel to the third side and equal to its half of the
82 trapezoid trapezoid theorem of the median line median line parallel to the two end and two equal to half of the bottom and L = (a + b) ÷ 2 S = L × h
83 (1) If the proportion of the basic properties of a: b = c: d, then ad = bc
if ad = bc, then a: b = c: d wc / S ?
84 (2) the nature of cooperation than if a / b = c / d, then (a ± b) / b = (c ± d) / d
85 (3) If the geometric properties of a / b = c / d = ... = m / n (b + d + ... + n ≠ 0), then
(a + c + ... + m) / (b + d + ... + n) = a / b
86 parallel line segments proportional to the theorem of three sub-parallel lines cut two lines from the corresponding line segments proportional reasoning
87 side of the line parallel to the triangle cut other side (or both sides of the extension cord), which is proportional to the corresponding segment < br> 88 Theorem If a line cut on both sides of the triangle (or both sides of the extension cord) from the corresponding line segments proportional, then this line parallel to the triangle's third side
89 parallel to the side of the triangle, and the other side intersecting lines, triangles intercepted by the triangular sides of a triangle corresponds to the original proportional
90 Theorem side of the line parallel to the triangle and the other side (or both sides of the extension line) intersects the triangle composed of the original triangle Similar
91 corners of similar triangles to determine the corresponding equivalent of Theorem 1, the two triangles similar to the (ASA)
92 high on the hypotenuse of a right triangle is divided into two right triangles and similar triangles of the original decision theorem 2
93 proportional to the corresponding sides and angles equal, the two triangles similar to the (SAS)
94 to determine the corresponding proportion trilateral Theorem 3, the two triangles similar to the (SSS)
95 Theorem If the hypotenuse of a right triangle and a right-angle edge hypotenuse of another right-angle side and a corresponding proportion, then the two triangles similar to the nature of Theorem 1
96 higher than the corresponding similar triangles, corresponding to the center line than the corresponding angle bisector is equal to the ratio nature than
97 similar similar triangles theorem 2 ratio is equal to the circumference of the nature similar to Theorem 3 than
98 similar similar triangles is equal to the area than the square
99 than the sine of any acute angle equal to its complementary angle of the cosine, any acute angle equal to its complementary angle cosine sine
100 tangent of any acute angle equal to its complementary angle cotangent value, the value of any acute angle equal to its complementary angle cotangent tangent of < br> 101 round is a distance equal to the fixed-length fixed-point set of points inside the circle
102 can be seen as the distance is less than the radius of the center point of the set
103 can be seen as outside the circle from the center of the circle greater than the radius of the set of points such
104 circle with the radius of a circle or equal
105 a distance equal to the fixed point of the trajectory of fixed length, is the point of a circle, a circle of radius length
106 and the distance between the two endpoints are known to line the locus of points equal to, is the perpendicular bisector of line segments to a known angle
107 equidistant points on both sides of the track, is the angle bisector
108-2 equidistant parallel lines locus of points, and it is these two parallel lines parallel and equidistant Theorem is not a straight line
109 three-point line has been identified with a circle.
110 vertical diameter perpendicular to the chord diameter of Theorem equally split this string by string and the two arcs
111 of Corollary 1 ① split string (not diameter) perpendicular to the string diameter, and equally the right chord arc
② two perpendicular bisector of string through the center of the circle, and the strings are split on the two strings of the arc
③ split an arc of diameter, the vertical split string, and split the string by another pair Corollary 2 arc
112 round clip of two parallel strings are equal
113 circle arc is the center of the center as the center of symmetry theorem of symmetry
114 or so in the same round in the circle, central angle equal The equivalent of the arc, the equivalent of the string, the string on the heart strings of inferences from the same
115 round or so in the same round, if the two central angle, two arcs, two string or two strings heart strings in a set amount of distance equal to that they correspond to the amount of the other groups are equal
116 theorem of an arc on the circumference of the angle of the central angle is equal to its half
117 Corollary 1 with the arc or other arc of the circumference of the equal angles; the same circle or other circular, the equivalent of the circumferential angle of the arc are equal
118 Corollary 2 semicircle (or diameter) of the angle of the circle at right angles; 90 ° circumferential angle The diameter of the string is
119 Corollary 3 If the center line on the side of the triangle is equal to half the side, then the triangle is right triangle within the circle theorems
120 quadrilateral diagonal complementary, and any exterior angle is is equal to its angle within
121 ① ⊙ O line L and intersects d
② ⊙ O line L, and d = r
③ tangent line L and ⊙ O phase from the d> r?
After 122 Tangent Theorem to determine the radius of the outer end and perpendicular to the radius of this circle tangent line is tangent to the nature of the Theorem
123 round after a cut perpendicular to the tangent point of radius
124 1 through the center of the circle and the inference will be perpendicular to the tangent line through the tangent point
125 Corollary 2 points after a cut perpendicular to the tangent line through the center
126 will be a long tangent point theorems introduced from outside the circle of the two tangent circles, their appearance and so the tangent , center, and split the connection that the angle between two tangents
127 quadrilateral circumscribed circle on the side of the two groups and equal
128 Xian Qiejiao Theorem Xian Qiejiao arc is equal to its folder circumferential angle
129 inference that if two Xianqie Jiao folder arc equal, then the two are equal
130 Xian Qiejiao intersection of the two strings intersect circle theorem of string, was divided into two intersection equal to the product of a long line
131 deduction if the chord and the diameter of vertical intersection, then the string by half its diameter into two sub-segments in the proportion of items
132 cutting line from the circle theorems point outside the circle tangent cited and the secant, tangent length is the secant and the circle to the intersection point of two line segments in the proportion of long-term
133 inferences that lead from outside the circle round the two secant, which is the secant and the circle of each the intersection of the product of two equal length segments if the two circles tangent
134, then the cutoff point must be in line with the heart outside the two circles
135 ① from the d> R + r ② d = exterior of two circles R + r
③ two circles intersect Rr r)
④ two circles inscribed d = Rr (R> r) ⑤ two circles containing d r)
136 two-circle theorem of intersecting lines with the axis of two circles the heart of public * String
137 into the circle theorem of n (n ≥ 3):
⑴ points followed by links from each polygon is The circle inscribed n-gon
⑵ points for each round after the tangent to the intersection of the tangent to the adjacent vertices of this polygon is a circle circumscribed n-gon theorem of any regular polygon
138 has a circumcircle and an inscribed circle, the two circles are concentric
139 regular n-gon of each interior angle is equal to (n-2) × 180 ° / n
140 theorems are n edge Heart-shaped edge radius and n from the regular 2n-gon into two congruent right triangle
141 n-gon is the area of Sn = pnrn / 2 p n-gon that is the perimeter of an equilateral triangle
142 area of √ 3a / 4 a side said
143 around if a vertex k-regular n-gon angle, and these angles should be 360 °, so k × (n-2) 180 ° / n = 360 ° into the (n-2) (k-2) = 4
144 arc length formula: L = n Wu R/180
145 fan-shaped area formula: S fan = n Wu R ^ 2 / 360 = LR / 2
146 within the common tangent length = d-(Rr) grandfather tangent length = d-(R + r)
multiplication and factoring
a ^ 2-b ^ 2 = ( a + b) (ab)
a ^ 3 + b ^ 3 = (a + b) (a ^ 2-ab + b ^ 2)
a ^ 3-b ^ 3 = (ab (a ^ 2 + ab + b ^ 2)
triangle inequality | a + b | ≤ | a | + | b | | ab | ≤ | a | + | b | | a | ≤ b <=>-b ≤ a ≤ b
| ab | ≥ | a | - | b | - | a | ≤ a ≤ | a |
the solution of a quadratic equation-b + √ (b ^ 2-4ac) / 2a-b- √ (b ^ 2-4ac) / 2a
relationship between roots and coefficients X1 + X2 =- b / a X1 * X2 = c / a Note: Whyte Theorem
discriminant
b ^ 2 - 4ac = 0 Note: The equation has two equal real roots
b ^ 2-4ac> 0 Note: The equation has two unequal real roots
b ^ 2-4ac <0 Note: The equation has no real root, root
a conjugate trigonometric formulas
corners and formulas
sin (A + B) = sinAcosB + cosAsinB
sin (AB) = sinAcosB-sinBcosA
cos ( A + B) = cosAcosB-sinAsinB
cos (AB) = cosAcosB + sinAsinB
tan (A + B) = (tanA + tanB) / (1-tanAtanB)
tan (AB) = (tanA -tanB) / (1 + tanAtanB)
cot (A + B) = (cotAcotB-1) / (cotB + cotA)
cot (AB) = (cotAcotB +1) / (cotB-cotA).
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