I. Proof of parallel lines

1 There is only one straight line between two points. The shortest line segment between two points is 3. The same angle or the complementary angle of the same angle is equal. 4. The same angle or the complementary angle of the same angle is equal. 5. Only one straight line is perpendicular to the known straight line. 6. Among all the line segments connected with points on a straight line, the shortest parallel axiom of a vertical line segment passes through a point outside the straight line. There is only one straight line parallel to this straight line. If both lines are parallel to the third line, the two lines are parallel to each other. The isosceles angles are equal and the two straight lines are parallel to each other. 10, the internal angles are equal, and the two straight lines are parallel to each other. 1 1 is complementary to the inner corner of the side, and the two straight lines are parallel to each other. 13, two straight lines are parallel.

15 The sum of two sides of the theorem triangle is greater than the third side 16 It is inferred that the difference between two sides of the triangle is less than the third side 17 The sum of the internal angles of the triangle and the three internal angles of the theorem triangle is equal to the complement of two acute angles of a right triangle 180 18/the external angles of the triangle are equal to its two. Angular axiom (SAS) has two triangles with equal angles. The corner axiom (ASA) has two triangles with equal angles and their clamping edges. These two triangles have congruent 24 inferences (AAS). Two triangles corresponding to the opposite sides of two angles and one angle are congruent with 25 sides (SSS), two triangles corresponding to three sides are congruent with 26 hypotenuses, and right-angled side axiom (HL). Two right-angled triangles are congruent with one hypotenuse and one right-angled side. 3. Basic Theorem of Triangle

Theorem 27 1 The distance from a point on the bisector of an angle to both sides of the angle is equal. Theorem 2 of 28 goes to a point where both sides of an angle are equidistant. On the bisector of this angle, the bisector of 29 angles is the set of all points that are equidistant from both sides of this angle. The nature theorem of isosceles triangle 30 The two base angles of an isosceles triangle are equal (that is, equilateral and equilateral). 3 1 Inference 1 The bisector of the top angle of the isosceles triangle bisects the bottom and is perpendicular to the bisector of the top angle of the isosceles triangle with the bottom 32. The midline on the bottom edge coincides with the height on the bottom edge. Inference 3 All angles of an equilateral triangle are equal, and each angle is equal to 60 34 isosceles triangle. If a triangle has two equal angles, 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 36 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-angled side it faces is equal to half of the hypotenuse. The median line of the hypotenuse of a right triangle is equal to half of the hypotenuse. Theorem 39 A point on the vertical line of a line segment is equal to the distance between the two endpoints of this line segment. The inverse theorem and the point where the two endpoints of a line segment are equal. On the midline of this line segment, the midline of line segment 4 1 can be regarded as a set of all points with equal distance from both ends of the line segment. Theorem 42: Two graphs that are symmetrical about a straight line are congruent. Theorem 43: Two figures are symmetrical about a straight line, then the symmetry axis is the median vertical line 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 bisected vertically 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 sides A and B of a right triangle is equal to the square of hypotenuse C, that is, A 2+B 2 = C 2 47 Inverse Theorem of Pythagorean Theorem If the lengths of three sides of a triangle are related to A,

Theorem 48 The sum of quadrilateral inner angles is equal to 360 49 and the sum of quadrilateral outer angles is equal to 360 50. The sum of the internal angles of the polygon and Theorem n 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. V. parallelogram proof

52 parallelogram property theorem 1 parallelogram diagonal equality 53 parallelogram property theorem 2 parallelogram opposite sides are equal 54 inference parallel line segments sandwiched between two parallel lines are equal 55 parallelogram property theorem 3 parallelogram diagonal bisection 56 parallelogram judgment theorem 1 two groups of parallelograms with equal diagonals are parallelograms 57 parallelogram judgment theorem. 2 Two sets of parallelograms with equal opposite sides are parallelograms 58. Parallelograms with equal diagonal lines are parallelograms 59. Parallelograms with equal opposite sides are parallelograms 60. Rectangular property theorem 1 Rectangular property theorem 6 1 Rectangular property theorem 2. Rectangular diagonal lines are equal 62. Rectangular property theorem 1. A quadrilateral with three right angles is a rectangle 63. Rectangular decision theorem 2. A parallelogram with equal diagonal lines is a rectangle 64. Rhombic property theorem 1 Rhombic property theorem 65. Rhombic diagonal lines are perpendicular to each other, and each diagonal line bisects a set of diagonal lines 66. Rhombic area = half of diagonal product. That is, S=(a×b)÷2 67 rhombus decision theorem 1 A quadrilateral with four equal sides is a rhombus 68 rhombus decision theorem 2 A parallelogram with diagonal lines perpendicular to each other is a rhombus 69 square property theorem 1 The four corners of a square are right angles, and all four sides are equal to 70 square property theorem 2 Two diagonal lines of a square are equal and bisected vertically, and each diagonal line bisects a set of diagonal lines.

Proof of isosceles trapezoid of intransitive verbs

74 isosceles trapezoid property theorem The two angles of isosceles trapezoid on the same base are equal. 75 The two diagonals of isosceles trapezoid are equal. 76 Isoisosceles trapezoid determination theorem A trapezoid with two equal angles on the same base is isosceles trapezoid. 77 A trapezoid with equal diagonals is isosceles trapezoid. 78 A parallel line bisection theorem If a group of parallel lines tangent to a straight line are equal, then the line segments tangent to other straight lines are also equal. 79 Inference 1 Through a straight line parallel to the bottom of the trapezoid, the other waist 80 must be equally divided. Inference 2 Through a straight line parallel to the other side of a triangle, the third side 8 1 must be bisected. The center line of a triangle is parallel to the third side and equal to half of it. The center line of the trapezoid is parallel to the two bottom sides and is equal to half of the sum of the two bottom sides. L =(。

Seven. Similar triangles's Proof and Definition

86 parallel lines are divided into segments. Proportional theorem Three parallel lines cut two straight lines, and the corresponding 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 extension lines of both sides), and the corresponding line segment is proportional. Theorem 88 If the corresponding line segments obtained by cutting two sides (or extension lines of two sides) of a triangle are proportional, then the third side 89 of the triangle is parallel to one side of the triangle. And the three sides of the triangular cut are proportional to the three sides of the original triangle. Theorem 90 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 triangle judgment theorem 1 equivalence of two angles. Similarity between two triangles (ASA) 92 A right triangle divided by the height on the hypotenuse is divided into two right triangles. Similarity with the original triangle 93 Judgment Theorem 2. Two sides are proportional and the included angles are equal. Similarity between two triangles (SAS) 94 Judgment Theorem 3. Three sides are proportional. Similarity of two triangles (SSS) theorem 95. If the hypotenuse and right-angled side of a right-angled triangle are proportional to the hypotenuse and right-angled side of another right-angled triangle, then the two right-angled triangles are similar. 96 Property Theorem 1 similar triangles has a high ratio, and the ratio of the corresponding median line to the corresponding angular bisector is equal to the similarity ratio. 97 Property Theorem 2 The ratio of similar triangles perimeter is equal to similarity ratio. Property Theorem 3 The ratio of similar triangles area is equal to the square of similarity ratio.

Eight. Proof and definition of chord and circle

The sine value of any acute angle is equal to the cosine value of other angles, and the cosine value of any acute angle is equal to the sine value of other angles. 100 The tangent of any acute angle is equal to the cotangent of the other angles. The cotangent value of any acute angle is equal to the tangent value of other angles 10 1. A circle is a set of points whose distance from a fixed point is equal to the fixed length 102. The interior of a circle can be regarded as a set of points whose distance from the center of the circle is less than the radius 103. The outer circle of a circle can be regarded as a group of points whose distance from the center of the circle is greater than the radius 104. The radius of the same circle or the same circle is equal to 13. The locus of a point with a distance equal to a fixed length is the locus of a fixed length circle with a radius of 106 and two endpoints of a known line segment with the same distance, the locus of a point with the same distance from the middle vertical line of a line segment 107 to both sides of a known angle, and the locus of a bisector of this angle 108 to a point with the same distance between two parallel lines. 1 10 Vertical Diameter Theorem bisects the chord perpendicular to the chord diameter and bisects the two arcs opposite to the chord11inference 1 ① bisects the diameter (not the diameter) of the chord perpendicular to the chord, and the midpoints of the two arcs opposite to the chord pass through the center of the circle, and the two arcs opposite to the chord. The perpendicular bisecting chord and bisecting another arc 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. 1 14 Theorem In the same circle or an equal circle, equal central angles have equal arcs and equal chords. The distance between chords of a pair of chords is equal. 1 15 It is inferred that in the same circle or the same circle, if the distances between two central angles, two arcs, two chords or two chords are equal, the corresponding other components are equal. 1 16 Theorem: The circumferential angle of an arc is equal to half of its central angle. In the same circle or equal circle, the arc opposite to the equal circle angle is also equal. 1 18 infers that 2 semicircles (or diameters) are right angles; The chord subtended by the circumferential angle of 90 is 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 the diagonal complement of the inscribed quadrilateral of the right triangle 120 theorem circle. And any outer angle is equal to the intersection point of the inner diagonal line 12 1① and ⊙O D < R2, and the tangent judgment theorem of ⊙O D = R3 and ⊙O D > R 122 passes through the outer end of the radius, and the straight line perpendicular to this radius is the tangent of the circle. 438+024 Inference 1 A straight line passing through the center and perpendicular to the tangent must pass through the tangent point 125 Inference 2 A straight line passing through the tangent point and perpendicular to the tangent must pass through the center 126 The tangent length theorem leads to two tangents 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. The sum of two opposite sides of the circumscribed quadrangle of a circle is equal. The tangent angle theorem is equal to the circumferential angle of the arc pair it clamps. It is deduced that if the arcs sandwiched by two chord tangent angles are equal, then the two chord tangent angles are equal to the two intersecting chords in the chord theorem circle. The product of the length of two lines divided by the intersection is equal to 13 1. It is deduced that if the chord intersects the diameter vertically, then half of the chord is the tangent and secant of the circle, which is drawn by the middle term 132 according to the ratio of two line segments formed by a point outside the circle. The tangent length is the ratio of the lengths of two lines from this point to the intersection of the secant and the circle. 133 This item infers that two secant lines are drawn from a point outside the circle, and the product of the lengths of the two lines from this point to the intersection of each secant line and the circle is equal to 134. If two circles are tangent, then the tangent point must be on the line 135① two circles are tangent to D > R+R ② two circles are tangent to d=R+r ③ two circles intersect R-R < D+R (R > R) ④ two circles are inscribed with D = R-R (R > R) ⑤ two circles contain D < R. The chord 137 theorem divides a circle into n (n ≥ 3): (1) The polygon obtained by connecting points in turn is an inscribed regular N polygon of the circle; (1) The circle passes through the tangents of each point, and the polygon whose vertices are the intersections of adjacent tangents is an circumscribed regular N polygon of the circle. These two circles are concentric circles 139. Every inner angle of a regular N-polygon is equal to the radius and area of the regular N-polygon in theorem (n-2) × 180/N 140, where apome divides the regular N-polygon into 2n congruent right-angled triangles 14 1. A/4A means that the side length is 144. The formula for calculating arc length is L=n R/ 180 145. The formula of sector area is s sector = n r 2/360 = lr/2.

Multiplication and Factorization of Common Mathematical Formulas 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|. B | | A | | B | | A | | B | | A | | B | | A | | B | | A | | B | | A | | B | | A | | B | | A | | B | | A | | A | | B | |-b ≤ a ≤ b | a-b |≥| a |-b | -a |≤ a | The relationship between the solution of quadratic equation in one variable-B+√ (B2-4ac)/2A root and the coefficient x 1+.

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 there is a formula for the sum of the formulas of two angles of trigonometric function with a yoke.

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

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)

Sine Theorem a/sinA=b/sinB=c/sinC=2R Note: where R stands for the radius of the circumscribed circle of a triangle, and cosine theorem b2=a2+c2-2accosB Note: Angle B is the standard equation (x-a)2+(y-b)2=r2 containing the circle of side A and side C Note: (A, B).

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

Parabolic standard equation Y2 = 2px2 =-2px2 =-2py cylinder lateral area S=c*h=2pi*h cone lateral area S= 1/2*c*l=pi*r*l arc length formula l=a*r a is the radian number of the central angle r > 0° sector area formula s =/kloc-.