1. There is only one straight line between two points.
2. The shortest line segment between two points.
3. The complementary angles of the same angle or equal angle are equal.
4. The complementary angles of the same angle or equal angle are equal.
5. There is one and only one straight line perpendicular to the known straight line.
6. Of all the line segments connecting a point outside the straight line with points on the straight line, the vertical line segment is the shortest.
7. The axiom of parallelism passes through a point outside a straight line, and one and only one straight line is parallel to this straight line.
8. If two straight lines are parallel to the third straight line, the two straight lines are also parallel to each other.
9. The same angle is equal and two straight lines are parallel.
10. The internal dislocation angles are equal and the two straight lines are parallel.
1 1. The inner angles on the same side are complementary and the two straight lines are parallel.
12. Two straight lines are parallel and have the same angle.
13. Two straight lines are parallel and the internal dislocation angles are equal.
14. Two straight lines are parallel and complementary.
15. Theorem: The sum of two sides of a triangle is greater than the third side.
16. Inference: The difference between two sides of a triangle is smaller than the third side.
17. Theorem of the sum of the internal angles of a triangle: the sum of the three internal angles of a triangle is equal to 180.
18. Inference 1: The two acute angles of a right triangle are complementary.
19. Inference 2: One outer angle of a triangle is equal to the sum of two non-adjacent inner angles.
20. Inference 3: An outer angle of a triangle is greater than any inner angle that is not adjacent to it.
2 1. congruent triangles has equal sides and angles.
22. Angular axiom: Two triangles with two sides and their included angles are congruent.
23. Angular axiom: Two triangles have two angles, and their sides are congruent.
24. Inference: There are two angles, and the opposite side of one angle corresponds to the congruence of two triangles.
25. Edge axiom: two triangles correspond to three equal congruences.
26. Axiom of hypotenuse and right-angled edge: Two right-angled triangles with hypotenuse and right-angled edge are congruent.
My junior high school student (ID: SSZZB _ CZB)
27. Theorem 1: The distance from a point on the bisector of an angle to both sides of the angle is equal.
28. Theorem 2: Points with equal distances to both sides of an angle are on the bisector of this angle.
29. The bisector of an angle is a collection of all points with equal distance to both sides of the angle.
30. The property theorem of isosceles triangle: the two base angles of isosceles triangle are equal.
3 1. Inference 1: The bisector of the top angle of the isosceles triangle bisects the bottom and is perpendicular to the bottom.
32. Inference 2: The bisector of the top angle, the median line on the bottom and the height of the isosceles triangle coincide.
Inference 3: All angles of an equilateral triangle are equal, and each angle is equal to 60.
34. Judgment theorem of isosceles triangle: If the two angles of the triangle are equal, then the opposite sides of the two angles are also equal (equilateral).
35. 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.
37. In a right triangle, if an acute angle is equal to 30, then the right side it faces is equal to half of the hypotenuse.
38. The median line on the hypotenuse of a right triangle is equal to half of the hypotenuse.
39. Theorem The point on the vertical line of a line segment is equal to the distance between the two endpoints of this line segment.
40. Inverse Theorem: The point where the two endpoints of a line segment are equidistant is on the middle vertical line of this line segment.
4 1. The middle vertical line of a line segment can be regarded as the set of all points with the same distance at both ends of the line segment.
42. Theorem 1: Two figures symmetrical about a straight line are conformal.
43. Theorem 2: If two figures are symmetrical about a straight line, then the symmetry axis is the middle vertical line connecting the corresponding points.
Theorem 3: Two figures are symmetrical about a straight line. If their corresponding line segments or extension lines intersect, then the intersection point is on the axis of symmetry.
45. Inverse theorem: If the straight line connecting the corresponding points of two graphs is vertically bisected by the same straight line, then the two graphs are symmetrical about this straight line.
46. Pythagorean theorem: The sum of squares of two right angles A and B of a right triangle is equal to the square of hypotenuse C, that is, A+B = C.
47. Pythagorean Theorem Inverse Theorem: If three sides of a triangle have a relationship a+b=c, then this triangle is a right triangle.
48. Theorem: The sum of the internal angles of a quadrilateral is equal to 360.
49. The sum of the external angles of the quadrilateral is equal to 360.
50. Theorem The sum of the internal angles of a polygon and an N-sided polygon is equal to (n-2) × 180.
5 1. Inference: The sum of the external angles of any polygon is equal to 360.
52. parallelogram property theorem 1: the diagonals of parallelograms are equal.
53. parallelogram property theorem 2: the opposite sides of parallelogram are equal
54. Inference: The parallel segments sandwiched between two parallel lines are equal.
55. parallelogram property theorem 3: diagonal bisection of parallelogram.
56. parallelogram decision theorem 1: two groups of quadrilaterals with equal diagonals are parallelograms.
57. parallelogram judgment theorem 2: two groups of parallelograms with equal opposite sides are parallelograms.
58. parallelogram decision theorem 3: quadrilaterals with diagonal lines bisecting each other are parallelograms.
59. parallelogram judgment theorem 4: a group of parallelograms with equal opposite sides are parallelograms.
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: A parallelogram with equal diagonals is a rectangle.
64. Diamond property theorem 1: All four sides of a diamond are equal.
65. Diamond Property Theorem 2: Diagonal lines of the diamond are perpendicular to each other, and each diagonal line bisects 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 with mutually perpendicular diagonals are diamonds.
69. Theorem of Square Properties 1: All four corners of a square are right angles and all four sides are equal.
70. Theorem 2 of Square Properties: The two diagonals of a square are equal and vertically divided, and each diagonal bisects a set of diagonals.
7 1. Theorem 1: The congruence of two centrally symmetric graphs.
72. Theorem 2: For two graphs with symmetrical centers, the connecting lines of symmetrical points pass through the symmetrical center and are equally divided by the symmetrical center.
73. Inverse theorem: If a straight line connecting the corresponding points of two graphs passes through a certain point and is equally divided by the point, then the two graphs are symmetrical about the point.
74. Isosceles trapezoid property theorem: The two angles of isosceles trapezoid on the same base are equal.
75. The two diagonals of an isosceles trapezoid are equal.
76. isosceles trapezoid judgment theorem: two trapezoid with equal angles are isosceles trapezoid on the same bottom.
77. A trapezoid with equal diagonal lines is an isosceles trapezoid.
78. Theorem of Equal Segment of Parallel Lines: If a group of parallel lines have equal segments on a straight line, then the segments on other straight lines are also equal.
79. Inference 1: A straight line passing through the midpoint of one waist of a trapezoid and parallel to the bottom will bisect the other waist.
80. Inference 2: A straight line passing through the midpoint of one side of a triangle and parallel to the other side will bisect the third side.
8 1. midline theorem of triangle: the midline of triangle is parallel to the third side, which is equal to half of it.
82. Trapezoidal mean value theorem: the midline of the trapezoid is parallel to the two bottoms and equal to half of the sum of the two bottoms L = (a+b) ÷ 2S = L× H.
83. Basic properties of (1) ratio 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) Isometric Properties 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 line segments according to the proportionality theorem: three parallel lines cut two straight lines, and the corresponding line segments are proportional.
87. Inference: A straight line parallel to one side of a triangle intersects with the other two sides (or extension lines on both sides), and the corresponding line segments are proportional.
Theorem: If the corresponding line segments cut by two sides (or extension lines of two sides) of a triangle are proportional, the straight line is parallel to the third side of the triangle.
89. A straight line parallel to one side of a triangle and intersecting with the other two sides, the three sides of the cut triangle are directly proportional to the three sides of the original triangle.
90. Theorem A straight line parallel to one side of a triangle intersects the other two sides (or extension lines on both sides), and the triangle is similar to the original triangle.
9 1. similar triangles's decision theorem 1: Two angles are equal and two triangles are similar (ASA).
92. The right triangle is divided into two right triangles according to the height on the hypotenuse, which is similar to the original triangle.
93. Decision Theorem 2: Two sides are proportional and the included angle is equal, and two triangles are similar (SAS).
94. Decision Theorem 3: Three sides are proportional and two triangles are similar (SSS).
95. Theorem: If the hypotenuse and a right-angled side of a right-angled triangle are directly proportional to the hypotenuse and a right-angled side of another right-angled triangle, then the two right-angled triangles are similar.
96. Property Theorem 1: similar triangles corresponds to the height ratio, the ratio corresponding to the center line and the ratio corresponding to the angular bisector are all equal to the similarity ratio.
97. Theorem 2: The ratio of similar triangles perimeter is equal to the similarity ratio.
Theorem 3: similar triangles area ratio is equal to the square of similarity ratio.
99. The sine value of any acute angle is equal to the cosine value of the remaining angles, and the cosine value of any acute angle is equal to the sine value of the remaining angles.
100. The tangent of any acute angle is equal to the cotangent of the remaining angles, and the cotangent of any acute angle is equal to the tangent of the remaining angles.
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 collection of points whose center distance is less than the radius.
103. The outside of a circle can be regarded as a collection of points whose center distance is greater than the radius.
104. Same circle or equal circle has the same radius.
105. The distance to the fixed point is equal to the trajectory of the fixed-length point, which is a circle with the fixed point as the center and the fixed length as the radius.
106. It is known that the locus of the points with the same distance between the two ends of a line segment is the middle vertical line of the line segment.
107. The locus of points with equal distance to both sides of a known angle is the bisector of this angle.
108. The locus to the equidistant point of two parallel lines is a straight line parallel to these two equidistant parallel lines.
109. Theorem: Three points that are not on a straight line determine a straight line.
1 10. vertical diameter theorem: the diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite to the chord.
1 1 1. Inference 1:
(1) bisects the diameter (not the diameter) of the chord perpendicular to the chord and bisects the two arcs opposite to the chord.
(2) The perpendicular line of the chord passes through the center of the circle and bisects the two arcs opposite to the chord.
③ bisect the diameter of an arc opposite to the chord, bisect the chord vertically, and bisect another arc opposite to the chord.
1 12. Inference 2: The arcs sandwiched by two parallel chords of a circle are equal.
1 13. A circle is a centrally symmetric figure with the center of the circle as the center of symmetry.
1 14. Theorem: In the same circle or in the same circle, equal central angles have equal arcs, equal chords and equal chord center distances.
1 15. Inference: In the same circle or equal circle, if one set of quantities in two central angles, two arcs, two chords or the distance between two chords is equal, the corresponding other set of quantities is also equal.
1 16. Theorem: The angle of an arc is equal to half of its central angle.
1 17. Inference 1: The circumferential angles of the same arc or equal arc are equal; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.
1 18. Inference 2: The circumferential angle (or diameter) of a semicircle is a right angle; A chord with a circumferential angle of 90 is a 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 lines of inscribed quadrangles of a circle are complementary, and any external angle is equal to its internal angle.
12 1 .① lines l and ⊙O intersect with d r.
(2) the tangent of the straight line l, and ⊙ o d = r.
③ straight lines l and ⊙O are separated by d r.
122. The judgment theorem of tangent is that the outer end of the radius and the straight line perpendicular to this radius are the tangents of the circle.
123. Tangent theorem: the tangent 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: Two tangents leading from a point outside the circle are equal in length, and the connecting line between the center of the circle and this point bisects the included angle of the two tangents.
127. The sum of two opposite sides of the circumscribed quadrilateral of a circle is equal.
128. Chord-tangent angle theorem: Chord-tangent angle is equal to the circumferential angle of the arc pair it clamps.
129. Inference: If the arc enclosed by two chord tangent angles is equal, then the two chord tangent angles are also equal.
130. Intersecting chord theorem: the length of two intersecting chords in a circle divided by the product of the intersection point is equal.
13 1. Inference: If the chord intersects the diameter vertically, then half of the chord is the proportional median of the two line segments formed by dividing it by the diameter.
132. Secant theorem: the tangent and secant of a circle are drawn from a point outside the circle, and the length of the tangent is the median term of the ratio of the lengths of the two lines from this point to the intersection of the secant and the circle.
133. Inference: The product of two secant lines of a circle drawn from a point outside the circle to the intersection of each secant line and the circle is equal.
134. If two circles are tangent, then the tangent point must be on the line.
135.① The distance between two circles is d-r+r+r.
(2) circumscribed circle d d = r+r.
③ Two circles intersect R-r﹤d﹤R+r(R﹥r).
④ two circles are inscribed with d = R-R(R￠R)⑤ two circles contain d￠R-R(R￠R).
136. Theorem: The intersection line of two circles bisects the common chord of two circles vertically.
137. Theorem: Divide a circle into n (n ≥ 3);
(1) The polygon obtained by connecting the points in turn is the inscribed regular N polygon of this circle.
(2) The tangent of a circle passing through each point, and the polygon whose vertex is the intersection of adjacent tangents is the circumscribed regular N polygon of the circle.
Theorem: Any regular polygon has a circumscribed circle and an inscribed circle, which are concentric circles.
139. Each inner angle of a regular N-polygon is equal to (n-2) ×180/n.
140. Theorem: The radius and vertex of a regular N-polygon divide the regular N-polygon into 2n congruent right triangles.
14 1. The area of a regular N-polygon Sn = pnrn/2p 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, since the sum of these corners should be 360, then K× (n-2) 180/n = 360 becomes (n-2)(k-2)=4.
144. formula for calculating arc length: L = n ∏ R/ 180.
145. sector area formula: s sector = n ∏ R/360 = LR/2.
146. length of inner common tangent =d-(R-r) length of outer common tangent =d-(R+r)