Article 1: Line
1, congruence angles of the same angle or the same angle are equal.
2. There is one and only one straight line perpendicular to the known straight line.
There is only one straight line between two points.
The line segment between two points is the shortest.
5. The complementary angles of the same angle or equal angle are equal.
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 parallel axiom passes through a point outside the straight line, and there is only one straight line 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. Theorem The distance between the point on the vertical line of a line segment and the two endpoints of this line segment is equal.
10, the inverse theorem and the point where the distance between the two endpoints of a line segment is equal are on the vertical line of this line segment.
1 1. The perpendicular line of a line segment can be regarded as the set of all points with equal distance at both ends of the line segment.
12, theorem 1 Two graphs symmetric about a straight line are conformal.
13, Theorem 2 If two figures are symmetrical about a straight line, then the symmetry axis is the perpendicular line connecting the corresponding points.
14, Theorem 3 Two graphs 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.
15, Inverse Theorem If the straight line connecting the corresponding points of two graphs is bisected vertically by the same straight line, then the two graphs are symmetrical about this straight line.
Theorem of geometric formula in junior high school: angle
16, the same angle is equal, and two straight lines are parallel.
17, internal dislocation angles are equal, and two straight lines are parallel.
18, the internal angles on the same side are complementary, and the two straight lines are parallel.
19, two straight lines are parallel and have the same angle.
20, two straight lines are parallel, and the internal dislocation angles are equal.
2 1, two straight lines are parallel and complementary.
22. Theorem 1 The distance from the point on the bisector of the angle to both sides of the angle is equal.
23. Theorem 2 The point where two sides of an angle are at the same distance is on the bisector of this angle.
24. The bisector of an angle is the set of all points with equal distance to both sides of the angle.
Chapter 2: Triangle
25. The sum of two sides of a theorem triangle is greater than the third side.
26. It is inferred that the difference between two sides of a triangle is less than the third side.
27. The sum of the internal angles of a triangle is equal to 180.
28. The two acute angles of1right triangle are complementary inferences.
29. Inference 2 An outer angle of a triangle is equal to the sum of two non-adjacent inner angles.
30. Inference 3 The outer angle of a triangle is larger than any inner angle that is not adjacent to it.
3 1, Pythagorean theorem The sum of squares of two right angles 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 Pythagorean Theorem If the three sides of a triangle have a relationship a+b=c, then this triangle is a right triangle.
Chapter 3: isosceles triangle and right triangle
33. The essence of the isosceles triangle theorem The two base angles of an isosceles triangle are equal.
34. It is inferred that the bisector of the vertex of 1 isosceles triangle bisects the bottom and is perpendicular to the bottom.
35. The bisector of the top corner of the isosceles triangle, the median line on the bottom and the height coincide.
36. Inference 3 All angles of an equilateral triangle are equal, and each angle is equal to 60.
37. Decision theorem of isosceles triangle If a triangle has two equal angles, then the sides of the two angles are also equal (equilateral).
38. 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.
40. 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.
4 1. The median line of the hypotenuse of a right triangle is equal to half of the hypotenuse.
Article 4: Similarity, congruent triangles
42. Theorem A straight line parallel to one side of a triangle intersects with the other two sides (or extension lines on both sides), and the triangle formed is similar to the original triangle.
43. similar triangles's decision theorem 1 Two angles are equal and two triangles are similar (ASA)
44. Two right triangles divided by the height on the hypotenuse are similar to the original triangle.
45. Decision Theorem 2: Two sides are proportional and the included angle is equal, and two triangles are similar (SAS).
46. Decision Theorem 3 Three sides are proportional and two triangles are similar (SSS).
Theorem If the hypotenuse and a right-angled side of a right-angled triangle are proportional to the hypotenuse and a right-angled side of another right-angled triangle, then the two right-angled triangles are similar.
48. The property theorem 1 similar triangles corresponds to the height ratio, the ratio of the corresponding centerline and the ratio of the corresponding angular bisector are all equal to the similarity ratio.
49. Property Theorem 2 The ratio of similar triangles perimeter is equal to the similarity ratio.
50. Property Theorem 3 The ratio of similar triangles area is equal to the square of similarity ratio.
5 1. The angular axiom has two triangles with equal angles.
52. The angle axiom has two angles and two triangles with equal corresponding sides.
53. It is inferred that there are two angles, and the opposite side of one angle corresponds to the congruence of two triangles.
54. The axiom of edges is that two triangles have three corresponding equilateral sides.
55. The axiom of hypotenuse and right angle has the coincidence of hypotenuse and a right angle corresponding to two right triangle.
56. congruent triangles has equal sides and angles.
Chapter 5: Quadrilateral.
57. The sum of the internal angles of a quadrilateral is equal to 360 degrees.
58. The sum of the external angles of the quadrilateral is equal to 360.
59. Theorem The sum of the interior angles of a polygon is equal to (n-2) × 180.
60. It is inferred that the sum of the external angles of any polygon is equal to 360.
6 1, parallelogram property theorem 1 parallelogram diagonal is equal
62. parallelogram property theorem 2 The opposite sides of a parallelogram are equal
63. Inference that parallel segments sandwiched between two parallel lines are equal.
64. parallelogram property theorem 3 diagonal bisection of parallelogram.
65. parallelogram judgment theorem 1 Two groups of quadrangles with equal diagonals are parallelograms.
66. parallelogram decision theorem 2 Two groups of quadrilaterals with equal opposite sides are parallelograms.
67. Parallelogram Decision Theorem 3 Quadrilaterals whose diagonals are bisected are parallelograms.
68. parallelogram decision theorem 4 A set of parallelograms with equal opposite sides is a parallelogram.
Chapter 6: Rectangle
69. Theorem of Rectangular Properties 1 All four corners of a rectangle are right angles.
70. Rectangular property theorem The two diagonals of a rectangle are equal.
7 1, rectangle judgment theorem 1 A quadrilateral with three right angles is a rectangle.
72. Rectangular Decision Theorem 2 A parallelogram with equal diagonals is a rectangle.
Chapter 7: Diamonds
73. Diamond Property Theorem 1 All four sides of a diamond are equal
74. Diamond Property Theorem 2 Diagonal lines of diamonds are perpendicular to each other, and each diagonal line bisects a set of diagonal lines.
75, diamond area = half of the diagonal product, that is, S=(a×b)÷2.
76. The diamond decision theorem 1 A quadrilateral with four equilateral sides is a diamond.
77. Diamond Decision Theorem 2 Parallelograms with diagonal lines perpendicular to each other are diamonds.
Chapter 8: Square
78. Theorem of Square Properties 1 Four corners of a square are right angles and four sides are equal.
79. Theorem of Square Properties 2 The two diagonals of a square are equal and bisected vertically, and each diagonal bisects a set of diagonals.
80. Theorem 1 is congruent about two centrosymmetric graphs.
8 1, Theorem 2 For two graphs with symmetric centers, the connecting lines of symmetric points both pass through the symmetric center and are equally divided by the symmetric center.
82. 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.
Chapter 9: isosceles trapezoid
83, isosceles trapezoid property theorem isosceles trapezoid on the same bottom of the two angles are equal.
84. The two diagonals of an isosceles trapezoid are equal.
85. Isosceles Trapezoids Judgment Theorem Two isosceles trapeziums on the same bottom are isosceles trapeziums.
86. A trapezoid with equal diagonal lines is an isosceles trapezoid.
Chapter 10: Divide equally
87. 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.
88. Inference 1 passes through a straight line parallel to the trapezoid waist bottom, and the other waist will be equally divided.
89. 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.
90. The midline theorem of triangle is parallel to the third side and equal to half of the third side.
9 1, trapezoid midline theorem The trapezoid midline is parallel to the two bottoms and equal to half of the sum of the two bottoms L = (a+b) ÷ 2s = l× h.
92. Basic properties of (1) ratio If a:b=c:d, then ad=bc If ad=bc, then A: B = C: D.
93.(2) Combinatorial properties If a/b=c/d, then (A B)/B = (C D)/D.
94.(3) Isometric property If a/b=c/d=…=m/n(b+d+…+n≠0), then (A+C+…+M)/(B+D+…+N) = A/B.
95. Proportional theorem of parallel line segments Three parallel lines cut two straight lines, and the corresponding line segments are proportional.
96. It is inferred that a straight line parallel to one side of a triangle cuts the other two sides (or extension lines on both sides), and the corresponding line segments obtained are proportional.
97. Theorem If the corresponding line segments obtained by cutting two sides (or extension lines on both sides) of a triangle are proportional, then this straight line is parallel to the third side of the triangle.
98. For 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.
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 other angles, and the cotangent of any acute angle is equal to the tangent of other angles.
Article 11
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 the circle can be regarded as a collection of points whose center distance is greater than the radius.
104, same circle or same circle radius.
105. The trajectory of a point whose distance to a fixed point is equal to a fixed length 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 point where the two endpoints of a line segment are equidistant is the midline of the line segment.
107, it is known that the locus of points with equal distance on both sides of an angle is the bisector of this angle.
108, the locus to the equidistant points of two parallel lines is a straight line parallel to and equidistant from these two parallel lines.
109, the theorem determines a straight line at three points that are not on a straight line.
1 10, the vertical diameter theorem bisects the chord perpendicular to the diameter of the chord and bisects the two arcs opposite the chord.
1 1 1, inference 1 ① bisect the diameter of the chord perpendicular to the chord (not the diameter) and bisect 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, it is inferred that the arcs between 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, the isocentric angle has equal arc, chord and chord center distance.
1 15. It is inferred that in the same circle or the same 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 its central angle.
1 17, it is inferred that 1 the circumferential angles of the same arc or equivalent arc are equal; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.
1 18, it is inferred that the circumferential angle (or diameter) of the 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, it is proved that the diagonals of the inscribed quadrilateral of a circle are complementary, and any external angle is equal to its internal diagonal.
12 1, ① line l intersects with ⊙ O. dR2 line l is tangent to ⊙ O. D = R3 line l is separated from ⊙ O..
122, tangent judgment theorem The straight line passing through the outer end of the radius and perpendicular to this radius is the tangent of the circle.
123, the property theorem of tangent. 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, the tangent length theorem leads to two tangents of a circle from a point outside the circle, and their tangent lengths are equal. The 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 quadrangle of a circle is equal.
128, chord angle theorem chord angle is equal to the circumferential angle of the arc pair it clamps.
129. From this, it can be inferred that if the arc sandwiched between two chordal angles is equal, then the two chordal angles are also equal.
130, intersection chord theorem The product of the length of two intersecting chords divided by the intersection point in a circle is equal.
13 1. It is inferred that if the chord intersects the diameter vertically, then half of the chord is the proportional average of the two line segments formed by its divided diameter.
132, the tangent theorem leads to the tangent and secant of the circle from a point outside the circle, and the tangent length 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. It is inferred that the product of two secant lines 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 Perimeter of two circles d﹥R+r ② Perimeter of two circles d=R+r③ Intersection of two circles R-R-D+R (R-R) ④ Perimeter of two circles D = R-R (R-R).
Theorem 136 The intersection of two circles bisects the common chord of two circles vertically.
137, the theorem divides the 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 and each inner angle of a regular n-polygon is equal to (n-2) ×180/n.
140, theorem The radius of a regular N-polygon and apothem divide the regular N-polygon into 2n congruent right triangles.
14 1, and the area of the regular n-polygon Sn=pnrn/2 p represents the perimeter of the regular n-polygon.
142, and the area of a regular triangle √3a/4 a 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. Calculation formula of arc length: L=nπR/ 180.
145, sector area formula: s sector =nπR/360=LR/2.
146, inner common tangent length = d-(R-r) outer common tangent length = d-(R+r)