1, there is only one straight line between two points.

2. The line segment between two points is the shortest.

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 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. The same angle is equal, and two straight lines are parallel.

10, internal dislocation angles are equal, and 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, the sum of two sides of a theorem triangle is greater than the third side.

16, the difference between two sides of the inference triangle is smaller than the third side.

17, the sum of the internal angles of the triangle and the theorem triangle is equal to 180.

18, it is inferred that the two acute angles of 1 right triangle are complementary.

19, Inference 2 An outer angle of a triangle is equal to the sum of two inner angles that are not adjacent to it.

20. Inference 3 The outer angle of a triangle is larger than any inner angle that is not adjacent to it.

2 1, the corresponding edge of congruent triangles is equal to the corresponding angle.

22. The edge axiom (SAS) has two edges, and their included angle corresponds to the congruence of two triangles.

23. The corner axiom (ASA) has two corners and two triangles with equal corresponding sides.

24. Inference (AAS) has two angles, and the opposite side of one angle corresponds to the congruence of two triangles.

25. The side-by-side axiom (SSS) has the congruence of two triangles whose three sides correspond to each other.

26. Axiom of hypotenuse and right-angled side (HL) Two right-angled triangles with hypotenuse and a right-angled side are congruent.

27. Theorem 1 The distance from the point on the bisector of the angle to both sides of the angle is equal.

28. Theorem 2 The point where two sides of an angle are equidistant is on the bisector of this angle.

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

30, the nature theorem of isosceles triangle The two bottom angles of an isosceles triangle are equal (that is, equilateral angles)

3 1, inference 1 The bisector of the vertex of the isosceles triangle bisects the base and is perpendicular to the base.

32. The bisector of the top angle, the median line on the bottom edge and the height on the bottom edge of the isosceles triangle coincide with each other.

33. Inference 3 All angles of an equilateral triangle are equal, and each angle is equal to 60.

34. Decision theorem of isosceles triangle If a triangle has two equal angles, then the sides of the two angles are also equal (equal angles and equal sides).

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 midline of 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. The inverse theorem and the equidistant point between the two endpoints of a line segment are on the 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 graphs symmetric about a straight line are conformal.

43. Theorem 2 If two figures are symmetrical about a straight line, then the symmetry axis is the perpendicular line connecting the corresponding points.

44. 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-angled sides A and B of a right-angled triangle is equal to the square of hypotenuse C, that is, a2+b2=c2.

47. Inverse Theorem of Pythagorean Theorem If the three sides of a triangle A, B and C are related to a2+b2=c2, then this triangle is a right triangle.

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

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

50. Theorem The sum of the interior angles of a polygon 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 is equal

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

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

55, parallelogram property theorem 3 diagonal bisection of parallelogram.

56. parallelogram judgment theorem 1 Two groups of quadrilaterals with equal diagonals are parallelograms.

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

58. parallelogram decision theorem 3 The quadrilateral whose diagonals are bisected is a parallelogram.

59. parallelogram decision theorem 4 A set of parallelograms with equal opposite sides is a parallelogram.

60. Theorem of Rectangular Properties 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 diamonds are perpendicular to each other, and each diagonal line bisects a set of diagonal lines.

66, diamond area = half of the 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 diagonal lines perpendicular to each other are diamonds.

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

70. Theorem of Square Properties 2 The two diagonals of a square are equal and bisected vertically, and each diagonal bisects a set of diagonals.

7 1 and theorem 1 are congruent for two centrally symmetric graphs.

72. Theorem 2 For two graphs with symmetric centers, the connecting lines of symmetric points pass through the symmetric centers and are equally divided by the symmetric centers.

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 isosceles trapezoid on the same bottom of the two angles are equal.

75. The two diagonals of an isosceles trapezoid are equal.

76. Isosceles Trapezoids Decision Theorem Two isosceles trapeziums on the same bottom are isosceles trapeziums.

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 the same segment on a straight line, then the segments on other straight lines are the same.

79. Inference 1 passes through a straight line parallel to the trapezoid waist bottom, and the other waist will be equally divided.

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, the midline theorem of a triangle The midline of a triangle is parallel to the third side and equal to half of it.

82. The trapezoid midline theorem is parallel to the two bottoms and equal to half of the sum of the two bottoms L = (a+b) ÷ 2s = l× h.

83, (1) the basic nature of the ratio:

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

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

84, (2) the ratio nature:

If a/b=c/d, then (a b)/b = (c d)/d.

85, (3) equal ratio properties:

If a/b=c/d=…=m/n(b+d+…+n≠0),

Then (a+c+…+m)/(b+d+…+n) = a/b.

86. Proportional theorem of parallel line segments Three parallel lines cut two straight lines, and the corresponding line segments are proportional.

87. 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.

Theorem If the corresponding line segments obtained by cutting two sides (or extension lines of two sides) of a triangle are proportional, then this 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 triangle are proportional to the three sides of the original triangle.

Theorem A straight line parallel to one side of a triangle intersects the other two sides (or extension lines of both sides), and the triangle formed 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. Two right-angled triangles divided by the height on the hypotenuse are 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).

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.

96. 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.

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

98. Property Theorem 3 The ratio of similar triangles area 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 other angles, and the cotangent of any acute angle is equal to the tangent of other 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 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. Theorem Three points that are not on the same straight line determine a circle.

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, reasoning 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, 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 intersection point d of straight line l and ⊙O

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

③ Lines L and ⊙O are separated from each other d>r.

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 the circle from a point outside the circle. Their tangents have the same center, and the connecting line of 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 Two circles are separated from each other D>R+R.

(2) circumscribed circle d d = r+r.

(3) the intersection of two circles R-rr)

④ inscribed circle d = r-r (r >); r)

⑤ Two circles contain dr)

136, theorem The intersection line 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/2p represents the perimeter of the regular n-polygon.

142, and the regular triangle area √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. calculation formula of arc length: L = NR/ 180.

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

146, inner common tangent length =d-(R-r) outer common tangent length =d-(R+r)

Sine theorem a/sinA=b/sinB=c/sinC=2R.

Note: where r represents the radius of the circumscribed circle of the triangle.

Cosine theorem b2=a2+c2-2accosB

Note: Angle B is the included angle between side A and side C..

Formula classification formula expression

Multiplication and factorization A2-B2 = (a+b) (a-b) A3+B3 = (a+b) (A2-AB+B2) A3-B3 = (A-B (A2+AB+B2))

Trigonometric inequality | A+B |≤| A |+B||||| A-B|≤| A |+B || A |≤ B < = > -b≤a≤b

|a-b|≥|a|-|b|-|a|≤a≤|a|

The solution of the unary quadratic equation -b+√(b2-4ac)/2a-b-√(b2-4ac)/2a

The relationship between root and coefficient x1+x2 =-b/ax1* x2 = c/a Note: Vieta theorem.

discriminant

B2-4ac=0 Note: This equation has two equal real roots.

B2-4ac >0 Note: The equation has two unequal real roots.

B2-4ac & lt； Note: The equation has no real root, but a complex number of the yoke.

formulas of trigonometric functions

Two-angle sum formula

sin(A+B)= Sina cosb+cosAsinBsin(A-B)= Sina cosb-sinBcosA

cos(A+B)= cosa cosb-Sina sinb cos(A-B)= cosa cosb+Sina sinb

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 difference product

2 Sina cosb = sin(A+B)+sin(A-B)2 cosa sinb = sin(A+B)-sin(A-B)

2 cosa cosb = cos(A+B)-sin(A-B)-2 sinasinb = cos(A+B)-cos(A-B)

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

tanA+tanB = sin(A+B)/cosAcosBtanA-tanB = sin(A-B)/cosAcosB

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

The sum of the first n terms of some series

1+2+3+4+5+6+7+8+9+…+n = n(n+ 1)/2 1+3+5+7+9+ 1 1+ 13+ 15+…+(2n- 1)= N2

2+4+6+8+ 10+ 12+ 14+…+(2n)= n(n+ 1) 12+22+32+42+52+62+72+82+…+N2 = n(n+ 1)(2n+ 1)/6

13+23+33+43+53+63+…n3 = N2(n+ 1)2/4 1 * 2+2 * 3+3 * 4+4 * 5+5 * 6+6 * 7+…+n(n+ 1)= n(n+ 1)(n+2)/3

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..

The standard equation of a circle (x-a)2+(y-b)2=r2 Note: (A, B) is the center coordinate.

General equation of circle x2+y2+Dx+Ey+F=0 Note: D2+E2-4f > 0

Parabolic standard equation y2 = 2 pxy 2 =-2 pxy 2 = 2 pxy 2 =-2py.

Side area of straight prism S=c*h Side area of oblique prism s = c' * h

Lateral area of a regular pyramid s = 1/2c * h' lateral area of a regular prism s = 1/2 (c+c') h'

The lateral area of the frustum S = 1/2(c+c')l = pi(r+r)l The surface area of the ball S=4pi*r2.

Lateral area of cylinder S=c*h=2pi*h lateral area of cone s =1/2 * c * l = pi * r * l.

The arc length formula l=a*ra is the radian number r > of the central angle; 0 sector area formula s= 1/2*l*r

Conical volume formula V= 1/3*S*H Conical volume formula V= 1/3*pi*r2h

Oblique prism volume v = s' l Note: where s' is the straight cross-sectional area and l is the side length.

Cylinder volume formula V=s*h cylinder V=pi*r2h