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I want all theorems, definitions and properties of triangles in mathematics.
Theorem 15 The sum of two sides of a triangle is greater than the third side.

16 infers that the difference between two sides of a triangle is smaller than the third side.

The sum of the internal angles of 17 triangle is equal to 180.

18 infers that the two acute angles of 1 right triangle are complementary.

19 Inference 2 One outer angle of a triangle is equal to two non-adjacent outer angles.

Internal angle sum

Inference 3: An outer angle of a triangle is larger than any other outer angle and is not in phase with it.

Adjacent internal angle

2 1 congruent triangles has equal sides and angles.

The axiom of edges and corners (SAS) has two sides equal to their included angle.

Two triangles are congruent.

The corner axiom (ASA) has two corners corresponding to their sides.

The two triangles of the match.

Inference (AAS) has two angles, and the opposite sides of one angle are equal.

Two triangles are congruent.

The axiom of 25 sides (SSS) has two triangles, and their three sides correspond to each other.

suit

The hypotenuse and the axiom of right angles (HL) have hypotenuse corresponding to right angles.

Two equal right triangles are congruent.

Theorem 27 1 Distance from a point on the bisector of an angle to both sides of the angle

different

Theorem 2 for a point with equal distance on both sides of an angle, here

On the bisector of the angle

The bisector of an angle 29 is the set of all points with equal distance to both sides of the angle.

close

The property theorem of 30 isosceles triangle The two base angles of an isosceles triangle are in phase.

Equal (that is, equal sides and equal angles)

3 1 Inference 1 The bisector of the vertices of an isosceles triangle bisects the base and is vertical.

Go straight to the bottom

The bisector of the top angle, the middle line of the bottom edge and the bottom edge of an isosceles triangle.

These heights are consistent.

Inference 3 All angles of an equilateral triangle are equal, and each angle

Equal to 60 degrees

Judgement theorem of isosceles triangle with two angles.

Equal, then the sides of these two angles are also equal (equilateral)

Inference 1 A triangle with three equal angles is an equilateral triangle.

Inference 2 An isosceles triangle with an angle equal to 60 is an equilateral triangle.

corniform

In a right triangle, if an acute angle equals 30, then it

The right-angled side is equal to half of the hypotenuse.

The center line of the hypotenuse of a right triangle is equal to half of the hypotenuse.

The point on the perpendicular line of Theorem 39 and the two endpoints of this line segment.

The distance is equal.

The inverse theorem and the point where the distance between the two endpoints of a line segment is equal, here.

On the perpendicular bisector of the line segment.

The median vertical line of 4 1 line segment can be regarded as equal to the distance between the two ends of the line segment.

A collection of all points

Theorem 42 1 Two graphs symmetric about a line are conformal.

Theorem 2 If two graphs are symmetrical about a straight line, they are symmetrical.

The axis is the median perpendicular to the line connecting the corresponding points.

Theorem 3 Two graphs are symmetrical about a straight line if they correspond.

When line segments or extension lines intersect, the intersection point is on the axis of symmetry.

Inverse Theorem If the corresponding points of two graphs are connected by the same straight line.

Vertical division, then these two figures are symmetrical about this line.

Pythagorean Theorem The sum of squares of two right angles A and B of a right triangle, etc.

In the square of hypotenuse c, that is, A 2+B 2 = C 2.

47 Inverse Theorem of Pythagorean Theorem If the three sides of a triangle are A, B and C respectively,

The relationship A 2+B 2 = C 2, then this triangle is a right triangle.

The sum of the quadrilateral internal angles of Theorem 48 is equal to 360.

The sum of the external angles of the quadrilateral is equal to 360.

50 Theorem of the Sum of the Internal Angles of Polygons The sum of the internal angles of n polygons is equal to (n-2).

× 180

5 1 It is inferred that the sum of the external angles of any polygon is equal to 360.

52 parallelogram property theorem 1 parallelogram diagonal equality

53 parallelogram property theorem 2 The opposite sides of parallelogram are equal

It is inferred that the parallel segments sandwiched between two parallel lines are equal.

55 parallelogram property theorem 3 The diagonals of parallelograms are mutually flat.

minute

56 parallelogram judgment theorem 1 two groups of quadrilaterals with equal diagonals respectively

It is a parallelogram.

57 parallelogram decision theorem 2 Two groups of quadrilaterals with equal opposite sides.

It is a parallelogram.

58 parallelogram decision theorem 3 quadrilateral with diagonal bisector is

parallelogram

59 parallelogram decision theorem 4 A group of parallelograms with equal opposite sides.

It is a parallelogram.

60 Rectangle Property Theorem 1 All four corners of a rectangle are right angles.

6 1 rectangle property theorem 2 The diagonals of rectangles are equal

62 Rectangular Decision Theorem 1 A quadrilateral with three right angles is a rectangle.

63 Rectangular Decision Theorem 2 Parallelograms with equal diagonals are rectangles

64 diamond property theorem 1 all four sides of the diamond are equal.

65 Diamond Property Theorem 2 Diagonal lines of diamonds are perpendicular to each other, and each

Diagonal lines bisect a set of diagonal lines.

66 Diamond area = half of diagonal product, that is, S=(a×b)÷2.

67 diamond decision theorem 1 A quadrilateral with four equilateral sides is a diamond.

68 Diamond Decision Theorem 2 Parallelograms whose diagonals are perpendicular to each other are diamonds.

shape

69 Theorem of Square Properties 1 Four corners of a square are right angles, and there are four.

All parties are equal.

Theorem of 70 Square Properties 2 The two diagonals of a square are equal, and

Divide each other vertically, and each diagonal bisects a set of diagonal lines.

Theorem 7 1 1 is congruent with respect to two centrosymmetric graphs.

Theorem 2 For two graphs with central symmetry, the connecting lines of symmetrical points are all composed of

Passing through the center of symmetry and being divided into two by the center of symmetry.

Inverse Theorem If the line connecting the corresponding points of two graphs passes through a certain point.

And by this one.

If the point is split in two, then the two graphs are symmetrical about the point.

74 isosceles trapezoid property theorem Two angles of isosceles trapezoid on the same base.

(to) equal to ...

The two diagonals of an isosceles trapezoid are equal.

76 isosceles trapezoid judgment theorem Two ladders with equal angles on the same base

The shape is isosceles trapezoid.

A trapezoid with equal diagonal lines is an isosceles trapezoid.

Theorem of bisecting line segments by parallel lines If a group of parallel lines are on a straight line.

Cut the line segment at the top

Equal, then the line segments cut on other straight lines are also equal.

79 Inference 1 Through a straight line parallel to the trapezoid waist bottom, it must be

Divide the other waist equally

Inference 2 A straight line passing through the midpoint of one side of a triangle and parallel to the other side.

Ok, it must be split equally.

Trilaterality

The median line theorem of 8 1 triangle The median line of a triangle is parallel to the third side.

, equal to it

Half of

The midline theorem of trapezoid is parallel to the two bases, and

Equal to the sum of the two bottoms.

Half l = (a+b) ÷ 2s = l× h。

Basic properties of ratio 83 (1) If a:b=c:d, then ad=bc.

If ad=bc, then a: b = c: d.

84 (2) Combinatorial Properties If A/B = C/D, then (A B)/B = (C

d)/d

85 (3) Equal Ratio Property If A/B = C/D = … = M/N (B+D+…+N

0), then

(a+c+…+m)/(b+d+…+n)=a/b

86 parallel lines are divided into segments and proportional theorems. Three parallel lines cut two straight lines.

, the corresponding obtained.

The line segments are proportional.

It is inferred that a straight line parallel to one side of a triangle intersects with the other two sides (or sides).

The extension of an edge), which is proportional to the corresponding line segment.

Theorem 88 If a straight line cuts both sides of a triangle (or the extension lines of both sides)

Long line) is proportional to the corresponding line segment, then this straight line is parallel to.

The third side of a triangle

89 is parallel to one side of the triangle and intersects with the other two sides.

Line, the three sides of the cut triangle correspond to the three sides of the original triangle.

example

Theorem 90 is a straight line parallel to one side and the other two sides (or sides) of a triangle.

The extension lines of the sides intersect to form a triangle similar to the original triangle.

9 1 similar triangles's decision theorem 1 Two angles are equal and two triangles.

Similarity (ASA)

92 Right Triangle Two right triangles are divided by the height on the hypotenuse.

Similar to the original triangle

Decision Theorem 2: Two sides are proportional and the included angle is equal, two triangles.

Shape similarity

94 Decision Theorem 3 Three sides are proportional and two triangles are similar (SSS

)

Theorem 95 If the hypotenuse and right-angled side of a right-angled triangle

Another right angle three

The hypotenuse of an angle is proportional to a right angle, so these two straight lines

Similarity of angle triangle

96 Property Theorem 1 similar triangles corresponds to the height ratio and the median line.

The ratio is equal to the corresponding angle.

The ratio of dividing lines is equal to the similarity ratio.

97 Property Theorem 2 The ratio of similar triangles perimeter is equal to similarity ratio.

98 Property Theorem 3 similar triangles area ratio is equal to the level of similarity ratio.

square

The sine of any acute angle is equal to the cosine of the remaining angles.

Cosine value of acute angle, etc.

Sine value of other angles

100 The tangent of any acute angle is equal to the cotangent of the remaining angles, and any

Cotangent value of acute angle, etc.

Tangent value of its complementary angle

10 1 A circle is a set of points whose distance from a fixed point is equal to a fixed length.

102 The interior of a circle can be regarded as a point whose center is smaller than the radius.

gather

The outer circle of 103 circle can be regarded as the point whose center is farther than the radius.

gather

104 The radius of the same circle or equal circle is the same.

The distance from 105 to a fixed point is equal to the trajectory of a fixed-length point, which is based on this fixed point.

The center of the circle, the fixed length is half.

Diameter circle

106 and the locus of the point with the same distance between the two endpoints of the known line segment are

Verticality of line segment

bisector

The trajectory from 107 to the point with the same distance on both sides of the known angle is this angle.

The bisector of

The trajectory from 108 to the point with the same distance from two parallel lines is the same as these two lines.

Parallel lines are parallel and separate.

A straight line of equality

Theorem 109 Three points that are not on the same straight line determine a circle.

1 10 vertical diameter theorem bisects the chord perpendicular to its diameter, bisects the chord.

Two opposite arcs

1 1 1 inference 1 ① The diameter (not the diameter) of the bisector is perpendicular to the chord.

And split the two arcs opposite the chord in two.

(2) The perpendicular bisector of a chord passes through the center of the circle and bisects two opposite chords.

arc

(3) bisect the diameter of an arc opposite to the chord, bisect the chord vertically, and be flat.

Another arc into which a chord is divided.

1 12 Inference 2 The arcs sandwiched by two parallel chords of a circle are equal.

1 13 circle is a centrosymmetric figure with the center of the circle as the symmetry center.

Theorem 1 14 In the same circle or in the same circle, the phases of circular arcs with equal central angles.

Wait, right chord.

Equal, the chord center distance of the opposite chord is equal.

1 15 inference in the same circle or the same circle, if two central angles and two arcs

, two strings or two

The distance between chords of a chord has a set of equal quantities, so the other quantities correspond to each other.

The ingredients are all equal.

Theorem 1 16 The circumferential angle of an arc is equal to its central angle.

one half

1 17 Inference 1 The circumferential angles of the same arc or the same arc are equal; Homologous or equal

In a circle, equal circumferential angles face equal arcs.

1 18 Inference 2 The circumferential angle (or diameter) of a semicircle is a right angle; 90

circumferential angle

The chord on the right is the diameter.

1 19 Inference 3 If the median line of one side of a triangle is equal to half of this side,

Then this triangle is a right triangle.

120 Theorem Diagonal Complementarity of the inscribed quadrilateral of a circle, and any one of them

The outer angle is equal to it.

Internal diagonal of

12 1① the intersection of the straight line l and ⊙O is d < r.

(2) the tangent of the straight line l, and ⊙ o d = r.

③ lines l and ⊙O are separated by d > r.

The judgment theorem of 122 tangent passes through the outer end of radius and is perpendicular to this line.

The straight line of the radius is the tangent of the circle.

123 The property theorem of tangent line The tangent line of a circle is perpendicular to the radius passing through the tangent point.

124 Inference 1 A straight line passing through the center of the circle and perpendicular to the tangent must pass through the tangent point.

125 Inference 2 A straight line passing through the tangent and perpendicular to the tangent must pass through the center of the circle.

126 tangent length theorem leads to two tangents of a circle from a point outside the circle, and their

Tangent length, etc.

The line between the center of the circle and this point bisects the included angle between the two tangents.

127 The sum of two opposite sides of a circle's circumscribed quadrilateral is equal.

128 Chord Angle Theorem The chord angle is equal to the circumferential angle of the arc pair it clamps.

129 Inference If the arcs sandwiched between two chordal angles are equal, then these two

The chord angles are also equal.

130 theorem of intersecting chords Two intersecting chords in a circle are divided into two by the intersection point.

Product of line segment length

(to) equal to ...

13 1 Inference: If the chord intersects the diameter vertically, then half of the chord is it.

Divided by diameter

Proportional median of two line segments

132 tangent theorem derives the tangent and secant of a circle from a point outside the circle, tangent

How long should this point be cut?

The proportional average of the lengths of two straight lines at the intersection of a straight line and a circle.

133 Derive two secant lines leading to the circle from a point outside the circle, and this point reaches each line.

The product of the lengths of the two lines where the secant intersects the circle is equal.

134 If two circles are tangent, then the tangent point must be on the line.

135① perimeter of two circles D > R+R ② perimeter of two circles d = r+r.

③ the intersection of two circles r-r < d < r+r (r > r).

④ inscribed circle D = R-R (R > R) ⑤ two circles contain D < R-R (R > R).

Theorem 136 The intersection of two circles bisects the common chord of two circles vertically.

Theorem 137 divides a circle into n (n ≥ 3);

(1) The polygon obtained by connecting the points in turn is the inscribed positive n of this circle.

Edge shape

⑵ Make a tangent of a circle through all points, and take the intersection of adjacent tangents as the vertex.

The polygon of is the circumscribed regular n polygon of this circle.

Theorem 138 Any regular polygon has a circumscribed circle and an inscribed circle.

These two circles are concentric circles.

139 every inner angle of a regular n-polygon is equal to (n-2) ×180/n.

140 Theorem Radius and apothem divide a regular N-polygon into 2n.

Congruent right triangle

14 1 the area of the regular n polygon Sn = PNRN/2 P represents the perimeter of the regular n polygon.

142 The area of a regular triangle √ 3a/4a indicates the side length.

143 if there are k positive n corners around a vertex, because this

The sum of some angles should be

360, so k× (n-2) 180/n = 360 is changed to (n-2).

(k-2)=4

The formula for calculating the arc length of 144 is L = NR/ 180.

145 sector area formula: s sector =n r 2/360 = LR/2.

146 formula of trigonometric function with inner common tangent length = d-(R-r) outer common tangent length = d-(R+r).

Two-angle sum formula

sin(A+B)=sinAcosB+cosAsinB

sin(A-B)=sinAcosB-sinBcosA

cos(A+B)=cosAcosB-sinAsinB

cos(A-B)=cosAcosB+sinAsinB

tan(A+B)=(tanA+tanB)/( 1-tanA tanB)

tan(A-B)=(tanA-tanB)/( 1+tanA tanB)

ctg(A+B)=(ctgActgB- 1)/(ctg B+ctgA)

ctg(A-B)=(ctgActgB+ 1)/(ctg b-ctgA)

Double angle formula

tan2A = 2 tana/( 1-tan2A)ctg2A =(ctg2A- 1)/2c TGA

cos2a = cos2a-sin2a = 2 cos2a- 1 = 1-2 sin2a

half-angle formula

sin(A/2)=√(( 1-cosA)/2)

sin(A/2)=-√(( 1-cosA)/2)

cos(A/2)=√(( 1+cosA)/2)

cos(A/2)=-√(( 1+cosA)/2)

tan(A/2)=√(( 1-cosA)/(( 1+cosA))

tan(A/2)=-√(( 1-cosA)/(( 1+cosA))

ctg(A/2)=√(( 1+cosA)/(( 1-cosA))

ctg(A/2)=-√(( 1+cosA)/(( 1-cosA))

Sum and difference of products

2sinAcosB=sin(A+B)+sin(A-B)

2cosAsinB=sin(A+B)-sin(A-B)

2cosAcosB=cos(A+B)-sin(A-B)

-2sinAsinB=cos(A+B)-cos(A-B)

Sum difference product

sinA+sinB = 2 sin((A+B)/2)cos((A-B)/2

cosA+cosB = 2cos((A+B)/2)sin((A-B)/2)

tanA+tanB = sin(A+B)/cosa cosb tanA-tanB = sin(A-

B)/cosAcosB

ctgA+ctgBsin(A+B)/Sina sin B- ctgA+ctgBsin

(A+B)/sinAsinB

Sine theorem a/sinA=b/sinB=c/sinC=2R Note: where r

Represents the radius of the circumscribed circle of a triangle.

Cosine Theorem b2=a2+c2-2accosB Note: Angle B is the included angle between side A and side C..

Inductive formula

sin(-a)=-sin(a)

cos(-a)=cos(a)

sin(π/2-a)= cos(a)

cos(π/2-a)= sin(a)

sin(π/2+a)= cos(a)

cos(π/2+a)=-sin(a)

sin(π-a)= sin(a)

cos(π-a)=-cos(a)

sin(pi+a)=-sin(a)

cos(pi+a)=-cos(a)

tgA=tanA=sinA/cosA

General formula of trigonometric function

sin(a)=(2tan(a/2))/( 1+tan^2(a/2))

cos(a)=( 1-tan^2(a/2))/( 1+tan^2(a/2))

Tan (1) = (1)

Other formulas

A * sin (a)+b * cos (a) = sqrt (a2+B2) sin (a+c) [where tan(c)=b/a]

A * sin (a)-b * cos (a) = sqrt (a2+B2) cos (a-c) [where tan(c)=a/b]

1+sin(a)=(sin(a/2)+cos(a/2))^2

1-sin(a)=(sin(a/2)-cos(a/2))^2

Other non-critical trigonometric functions

csc(a)= 1/sin(a)

Seconds (a)= 1/ cosine (a)