【Junior High School Mathematics】The whole book of formula theorems is summarized, super useful!

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01

line

1. Co-angles of the same or equal angles are equal

2. There is only one and only one straight line perpendicular to the known straight line

3. There is only one straight line after two points

4. The shortest line segment between two points

5. The complementary angles of the same angle or equal angle are equal

6. Among all the line segments connected by a point outside the line and each point on the line, the perpendicular line segment is the shortest

7. The axiom of parallelism After a point outside the straight line, there is only one and only one straight line parallel to the straight line

8. If both lines are parallel to the third line, the two lines are also parallel to each other

9. Theorem The point on the perpendicular bisector of a line segment is equal to the distance between the two endpoints of this line segment

10. Inverse theorem and a point where two endpoints of a line segment are equally distanced, on the perpendicular bisector of this line segment

11. The perpendicular bisector of a line segment can be seen as a set of all points at equal distances from the two ends of the line segment

12. Theorem 1 Two figures about the symmetry of a certain straight line are congruent

13. Theorem 2 If two graphs are symmetrical with respect to a straight line, then the axis of symmetry is the perpendicular bisector of the corresponding point line

14. Theorem 3 Two graphs are symmetrical about a certain straight line, and if their corresponding segments or extensions intersect, then the intersection is on the axis of symmetry

15. Inverse theorem If the corresponding points of two graphs are bisected perpendicular to the same straight line, then the two graphs are symmetrical with respect to that straight line

02

horn

16. The isotope angles are equal, and the two straight lines are parallel

17. The inner wrong angles are equal, and the two straight lines are parallel

18. The inner angles of the same side are complementary, and the two straight lines are parallel

19. Two straight lines are parallel and the isotope angles are equal

20. The two straight lines are parallel, and the internal wrong angles are equal

21. The two straight lines are parallel and complementary to the side inner angles

22. Theorem 1 The distance from a point on the bisector of an angle to both sides of this angle is equal

23. Theorem 2 A point at the same distance to both sides of an angle, on the bisector of this angle

24. The bisector of an angle is the set of all points at equal distances to both sides of the angle

03

triangle

25. Theorem The sum of the two sides of a triangle is greater than the third side

26. Corollary The difference between the two sides of the triangle is less than the third side

27. Triangle Inner Angles Sum Theorem The sum of the three interior angles of a triangle is equal to 180°

28. Corollary 1 The two acute angles of a right triangle are surplus to each other

29. Corollary 2 One outer angle of a triangle is equal to the sum of two inner angles that are not adjacent to it

30. Corollary 3 One outer angle of a triangle is greater than any one of its inner angles that are not adjacent to it

31. Pythagorean theorem The sum of the squares of two right-angled sides A and B of a right triangle is equal to the square of the hypotenuse c, that is, a+b=c

32. Inverse theorem of the Pythagorean theorem If the three sides of a triangle are related to a, b, and c a+b=c, then the triangle is a right-angled triangle

04

Isosceles, right-angled triangle

33. Theorem of properties of isosceles triangle The two base angles of an isosceles triangle are equal

34. Corollary 1 The bisector of the apex angle of an isosceles triangle bisects the base edge and is perpendicular to the base edge

35. The bisector of the apex of an isosceles triangle, the midline on the bottom edge, and the height coincide with each other

36. Corollary 3 The angles of an equilateral triangle are equal, and each angle is equal to 60°

37. Determination theorem of isosceles triangle If a triangle has two equal angles, then the sides opposite the two angles are also equal (equigonal to equilateral)

38. Corollary 1 A triangle with three equal angles is an equilateral triangle

39. Corollary 2 An isosceles triangle with an angle equal to 60° is an equilateral triangle

40. In a right-angled triangle, if an acute angle is equal to 30°, then the right-angled side it is facing is equal to half of the hypotenuse

41. The middle line on the hypotenuse of a right triangle is equal to half of the hypotenuse

05

Similar, congruent triangles

42. Theorem A straight line parallel to one side of a triangle intersects with the other two sides (or extensions on both sides) to form a triangle similar to the original triangle

43. Similar triangle determination theorem 1 Two angles correspond to equal, two triangles are similar (ASA)

44. The right triangle is divided into two right triangles by the high division on the hypotenuse and the original triangle is similar to the original triangle

45. Decision theorem 2 The two sides correspond proportionally and the angles are equal, and the two triangles are similar (SAS)

46. Decision Theorem 3 Three sides correspond proportionally, two triangles are similar (SSS)

47. Theorem If the hypotenuse and one right-angled side of a right-angled triangle correspond to the hypotenuse and a right-angled side of another right-angled triangle, then the two right-angled triangles are similar

48. Property Theorem 1 The ratio of similar triangles corresponds to the height, and the ratio of the corresponding midline and the ratio of the bisector of the corresponding angle are all equal to the similarity ratio

49. Property Theorem 2 The ratio of the circumference of similar triangles is equal to the similarity ratio

50. Property Theorem 3 The ratio of the area of a similar triangle is equal to the square of the similarity ratio

51. The axiom of corner edges There are two triangles with equal sides and their angles corresponding to congruence

52. Corner Side Angle Axiom There are two angles and their intersections corresponding to two triangles that are congruent

53. Corollary There are two angles and two triangles that correspond to equal congruence on the opposite side of one of the corners

54. The edge edge side axiom There are three sides corresponding to two equal triangle congruence

55. Axiom of hypotenuse and right-angled edge There are two right-angled triangles that have an hypotenuse and a right-angled side corresponding to the equal congruence

56. The corresponding sides and corresponding angles of congruent triangles are equal

06

quadrilateral

57. Theorem The sum of the internal angles of a quadrilateral is equal to 360°

58. The sum of the outer angles of the quadrilateral is equal to 360°

59.多边形内角和定理 n边形的内角的和等于(n-2)×180°

60. Corollary The sum of the outer angles of any multilateral is equal to 360°

61. Parallelogram property theorem 1 The diagonal of a parallelogram is equal

62. Parallelogram property theorem 2 The opposite sides of a parallelogram are equal

63. Corollary Parallel segments sandwiched between two parallel lines are equal

64. Parallelogram properties theorem 3 The diagonal lines of parallelograms are bisected from each other

65. Parallelogram Determination Theorem 1 Two sets of quadrilaterals with equal diagonal angles are parallelograms

66. Parallelogram Decision Theorem 2 Two sets of quadrilaterals with equal opposite sides are parallelograms

67. Parallelogram Decision Theorem 3 A quadrilateral with diagonals bisecting each other is a parallelogram

68. Parallelogram Decision Theorem 4 A set of quadrilaterals with parallel and equal opposite sides is a parallelogram junior geometric formula theorem: rectangle

69. Rectangle property theorem 1 The four corners of a rectangle are all right angles

70. Theorem of rectangular properties 2 The diagonal lines of rectangles are equal

71. Rectangle Determination Theorem 1 There are three quadrilaterals with right angles that are rectangular

72. Rectangle Determination Theorem 2 A parallelogram with equal diagonals is a rectangle junior geometric formula: a diamond

73. Rhomboid Properties Theorem 1 The four sides of a rhomboid are all equal

74. Rhomboid properties theorem 2 The diagonals of a rhombus are perpendicular to each other, and each diagonal is bisected by a set of diagonals

75. Diamond area = half of the diagonal product, i.e. S = (a×b) ÷2

76. Rhombic Determination Theorem 1 A quadrilateral with all four sides equal is a rhombus

77. Rhomboid Determination Theorem 2 A parallelogram with diagonal perpendicular lines to each other is a rhombus

07

square

78. Theorem of the properties of the square 1 The four corners of a square are all right angles, and all four sides are equal

79. Square property theorem 2 The two diagonals of a square are equal and bisected perpendicular to each other, with each diagonal bisecting a set of diagonals

80. Theorem 1 Two figures with respect to central symmetry are congruent

81. Theorem 2 With regard to two figures of central symmetry, the lines of symmetry points pass through the center of symmetry and are bisected by the center of symmetry

82. Inverse theorem If the corresponding points of two graphs pass through a certain point and are bisected by this point, then the two graphs are symmetrical with respect to this point

08

Isosceles trapezoidal

83. Isosceles trapezoidal property theorem The two angles of an isosceles trapezoid are equal on the same base

84. The two diagonals of an isosceles trapezoidal are equal

85. Isosceles trapezoidal determination theorem Two trapezoids with equal angles on the same base are isosceles trapezoids

86. A trapezoid with equal diagonal lines is an isosceles trapezoid

09

Equal

87. Parallel Line Equalization Line Segment Theorem If a group of parallel lines is equal in the segments of a straight line, then the segments are also equal in the other lines

88. Corollary 1 A straight line that passes through the midpoint of a trapezoidal waist parallel to the bottom will divide the other waist equally

89. Corollary 2 A straight line that passes through the midpoint of one side of the triangle and parallel to the other side must be bisected by the third side

90. Median line theorem of triangles The median line of a triangle is parallel to the third side and is equal to half of it

91. Trapezoidal median line theorem The median line of the trapezoid is parallel to the two bases and is equal to half of the sum of the two bases L=(a+b)÷2 S=L×h92 ,

92. Basic properties of proportionality If a:b=c:d, then ad=bc If ad=bc, then a:b=c:d93

93.合比性质 如果a/b=c/d,那么(a±b)/b=(c±d)/d94

94.等比性质 如果a/b=c/d=…=m/n(b+d+…+n≠0)，那么，(a+c+…+m)/(b+d+…+n)=a/b

95. Parallel line division segment proportionality theorem Three parallel lines cut two straight lines, and the corresponding line segments obtained are proportional

96. Inference A straight line parallel to one side of the triangle cuts off the other two sides (or extensions on both sides), and the resulting corresponding line segments are proportional

97. Theorem If a straight line is proportional to the corresponding segments obtained by cutting off the two sides of a triangle (or the extension of both sides), then the line is parallel to the third side of the triangle

98. A straight line that is parallel to one side of the triangle and intersects the other two sides, the three sides of the truncated triangle are proportional to the three sides of the original triangle

99. The sine of any acute angle is equal to the cosine of its coangle, and the cosine of any acute angle is equal to the sine of its coangle

100. The tangent of any acute angle is equal to the cotangent of its coangle, and the cotangent of any acute angle is equal to the tangent of its coangle

10

round

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

102. The inside of a circle can be seen as a collection of points whose distance from the center of the circle is less than the radius

103. The outer part of a circle can be seen as a collection of points whose distance from the center of the circle is greater than the radius

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

105. The trajectory of a point whose distance to a fixed point is equal to its fixed length is a circle with a fixed point as the center and a fixed length as the radius

106. The trajectory of a point at a distance equal to the distance between two endpoints of a known line segment is the perpendicular bisector of a line segment

107. The trajectory to a point of equal distance on both sides of a known angle is the bisector of this angle

108. A trajectory to a point with equal distances from two parallel lines is a straight line parallel to these two parallel lines at equal distances

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

110. Perpendicular diameter theorem Bisect the string perpendicular to the diameter of the string and bisect the two arcs opposite the string

111. Corollary 1

(1) The diameter of the bisector chord (not the diameter) is perpendicular to the chord, and the two arcs of the bisector chord are opposed

(2) The perpendicular bisector of the string passes through the center of the circle and bisects the two arcs opposite the chord

(3) The diameter of one arc to which the bisector chord is paired, perpendicular to the bisector chord, and the other arc to which the bisector chord is opposed

112. Corollary 2 The arcs sandwiched by the two parallel strings of a circle are equal

113. A circle is a central symmetrical figure with the center of the circle as the center of symmetry

114. Theorem In the same circle or equal circle, the arcs opposite to the central angles of equal circles are equal, the opposite strings are equal, and the chord centric distance of the paired strings is equal

115. Corollary In the same circle or equal circle, if one set of quantities is equal in the central distance of two circles, two arcs, two strings, or two chords, then the rest of the groups of quantities to which they correspond are equal

116. Theorem The circumferential angle of an arc is equal to half of the central angle of the circle to which it is opposed

117. Corollary 1 The circumferential angles of the same arc or equal arc are equal; In the same circle or equal circle, the arcs opposite the circumferential angles of the same circle are also equal

118. Corollary 2 The circumferential angle of the semicircle (or diameter) is a right angle; The chord to which the circumferential angle of 90° is aligned is the diameter

119. Corollary 3 If the middle line on one side of a triangle is equal to half of this side, then the triangle is a right triangle

120. Theorem The diagonal complementarity of the inner quadrilateral of a circle is complementary, and any one outer angle is equal to its inner diagonal

121.

①直线L和⊙O相交 d﹤r

②直线L和⊙O相切 d=r

(3) The straight line L and ⊙O are separated by d>r

122. Decision theorem of tangents A straight line that passes through the outer end of a radius and is perpendicular to this radius is a tangent of a circle

123. Theorem of the nature of tangent lines The tangent of a circle is perpendicular to the radius passing through the tangent point

124. Corollary 1 A straight line that passes through the center of a circle and is perpendicular to the tangent must pass through the tangent point

125. Corollary 2 A straight line that passes through a tangent and is perpendicular to the tangent must pass through the center of the circle

126. Tangent length theorem Two tangents that lead a circle from a point outside the circle are equal in length, and the center of the circle and the line connecting this point are equally divided between the angles of the two tangents

127. The sum of the two sets of opposite sides of the circumscribed quadrilateral of the circle is equal

128. Chord chamfer theorem The chord chamfer is equal to the circumferential angle of the pair of arcs it is sandwiched

129. Corollary If the arcs sandwiched by two chord tangent angles are equal, then the two chord tangent angles are also equal

130. Intersecting string theorem Two intersecting strings in a circle are divided by the intersection point into which the product of the length of the two line segments is equal

131. Corollary If the string intersects perpendicular to the diameter, then half of the string is the proportional term of the two segments of the line into which it divides the diameter

132. The tangent line theorem The tangent and secant line of the circle from a point outside the circle, the tangent length is the middle term of the ratio between this point and the length of the two line segments at the intersection of the secant line and the circle

133. Corollary From the point outside the circle lead the two secant lines of the circle, to this point that the product of the length of the two segments of each secant is equal to the intersection of the circle

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

135.

(1) The two circles are separated from d>R+r

(2) The two circles are inscribed d=R+r

③两圆相交 R-r﹤d﹤R+r(R﹥r)

④两圆内切 d=R-r(R﹥r) ⑤两圆内含d﹤R-r(R﹥r)

136. Theorem The concentric line that intersects two circles is perpendicular to the common chord of the two circles

137. Theorem Divide a circle into n(n≥3):(1) The polygon obtained by connecting the points in turn is the inner regular n-sided of the circle, (2) the tangent of the circle is made through the points, and the polygon with the intersection of adjacent tangents as the vertex is the outer tangent n-sided of the circle

138. Theorem Any regular polygon has an inscribed circle and an inscribed circle, and these two circles are concentric circles

139.正n边形的每个内角都等于(n-2)×180°/n

140. Theorem The radius and centroid distance of a regular n-sided divide a regular n-sided into 2n congruent right-angled triangles

141. The area of a regular n-sided Sn=pnrn/2 p denotes the circumference of a regular n-side

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

143. If there are k regular n-sided angles around a vertex, since the sum of these angles should be 360°, k×(n-2)180°/n=360° becomes (n-2)(k-2)=4

144. The formula for calculating arc length: L=nπR/180

145.扇形面积公式：S扇形=nπR/360=LR/2

146.内公切线长= d-(R-r) 外公切线长= d-(R+r）

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