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 offset 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. The internal dislocation angle is equal to 14, and the two straight lines are parallel. Theorem The sum of two sides of a triangle is greater than the third side 15. The difference between two sides of the reasoning triangle is less than the third side 17. Theorem The sum of three angles of a triangle is equal to 180 18. The two acute angles of a right triangle complement each other 19. The sum of the two angles of a triangle is equal to 18. The outer angle is equal to the sum of two non-adjacent inner angles. 20 Inference 3 An outer angle of a triangle is larger than the corresponding side of any inner angle that is not adjacent to it, and 2 1 congruent triangles, and the corresponding angles are equal. 22-Angle Axiom (SAS) has two triangles with equal included angles (ASA). Two angles of two triangles correspond to their sides congruent 24 Inference (AAS) Two angles of two triangles correspond to one opposite side congruent 25-sided axiom (SSS) Two triangles correspond to congruent 26 hypotenuse and right-angled axiom (HL). Two right-angled triangles with a hypotenuse and a right-angled side are congruent. Theorem 1 A point on the bisector of an angle is equal to the distance between two sides of the angle. Theorem 2 To a point with equal distance on both sides of an angle, on the bisector of this angle, the bisector of 29 angles is the set of all points with equal distance on 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-angled sides A and B of a right-angled triangle is equal to the square of the hypotenuse C, that is, the inverse theorem of A 2+B 2 = C 2. 47 Pythagorean Theorem If three sides of a triangle have a relationship A 2+B 2 = C 2, then this triangle is a right-angled triangle. Theorem 48 The sum of internal angles of quadrilateral is equal to 360 49, and the sum of internal angles of polygon is equal to 360 50. Theorem n The sum of the internal angles of a polygon is equal to (n-2) × 180 5 1. It is inferred that the sum of external angles of any polygon is equal to 360 52, and the diagonal of parallelogram is equal to 53. Parallelogram property theorem 2 Parallelogram with equal opposite sides 54 Parallelogram property theorem 3 Parallelogram diagonal bisection 56 Parallelogram decision theorem 1 Two sets of parallelograms with equal diagonal are parallelograms 57 Parallelograms decision theorem 2 Two sets of parallelograms with equal opposite sides are parallelograms 58 parallel sides. Shape Decision Theorem 3 The quadrilateral whose diagonal is bisected is a parallelogram 59. Parallelogram Decision Theorem 4 A group of parallelograms whose opposite sides are parallel and equal is a parallelogram 60. Rectangular property theorem 1 rectangular property theorem 6 1 rectangular property theorem 2. Rectangular diagonal is equal to 62. Rectangular decision theorem 1 a quadrilateral with three right angles is a rectangle 63. Theorem 2 A parallelogram with equal diagonals is a rectangle 64. The four sides of the diamond 1 are equal. 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 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 All 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 divided vertically. Each diagonal bisects a set of diagonals 7 1 theorem 1 congruence of two figures symmetrical about the center 72 Theorem 2 For two figures symmetrical about the center, the straight line of the symmetrical point passes through the symmetrical center and is bisected by the symmetrical center 73 Inverse Theorem If the straight line of the corresponding point of two figures passes through a point and is bisected by the point, the two figures are symmetrical about the point. Property theorem of isosceles trapezoid. The two angles of an isosceles trapezoid on the same base are equal. The two diagonals of an isosceles trapezoid are equal. 76 isosceles trapeziums have equal angles on the same base, which is an isosceles trapezoid. The diagonal trapezoid is an isosceles trapezoid. Theorem of bisecting line segments by 78 parallel lines. If a set of parallel lines cut on a straight line are equal, then the line segments cut on 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 Inference 2 Through a straight line parallel to the other side of the triangle, the third side must be bisected. 8 1 The midline theorem of the triangle is parallel to the third side. And equal to half of it. The trapezium midline theorem is parallel to the two bottoms and is equal to half the sum of the two bottoms. Basic properties of L=(a+b)÷2 S=L×h 83 (1) If a:b=c:d, then ad=bc If ad=bc, then A: B = C: D. Then (A B)/B = (C D)/D. Then the corresponding line segment obtained by (A+C+…+M)/(B+D) infers that the straight line parallel to one side of the triangle cuts the other two sides (or the extension lines on both sides), and the corresponding line segment obtained is proportional to Theorem 88. If the corresponding line segments obtained by cutting two sides (or extension lines of two sides) of a triangle with a straight line are proportional, then this straight line is parallel to the third side 89 of the triangle, parallel to one side of the triangle and intersects with the other two sides. The three sides of the cutting triangle correspond to the three sides of the original triangle in proportion. 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. Theorem 1 similar triangle judgment theorem 1 two angles are equal. 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. Two triangles are similar (SSS) Theorem 95 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. Theorem 1 similar triangles corresponding height ratio. The ratio of the corresponding median line to the bisector of the corresponding angle is 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 the 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 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 1. 05 is the locus of a fixed-length point, the locus of a circle with a fixed length of half diameter 106 with the same distance from the two endpoints of a known line segment, the locus of a point with the same distance from the midline of the line segment 107 to both sides of a known angle, and the locus of a point with the same distance from the bisector of the angle 108 to two parallel lines. The vertical diameter theorem parallel to 1 10 bisects the chord perpendicular to the chord diameter and bisects the two arcs opposite to the chord.11inference 1 ① bisects the diameter (not the diameter) of the chord perpendicular to the chord, and the middle perpendicular of the two arcs opposite to the chord passes through the center of the circle and is 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 R 122 passes through the outer end of the radius, and the straight line perpendicular to this radius is the tangent of the circle. Radius 124 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 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. 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. The area √ 3a/4a indicates that the side length is 143. If there are K positive N corners around a vertex, since the sum of these angles should be 360, k × (n-2) 180/n = 360 is converted into (n-2)(k-2)=4 144. Arc length calculation formula: L=n R/ 180 145. Sector area formula: S.A3+B3 = (a+b) (A2-AB+B2) A3-B3 = (A-B (A2+AB+B2) Trigonometric Inequality | A+B |≤| A |+B | | A-B |≤|||||||||||||||||||||||||||||||||||||||| ||||||||||||||||| ||| | | | -b |-a |≤ a | The solution of a quadratic equation in one variable -b+√(B2-4ac)/2a-b-√(B2-4ac)/2a root and the relationship with the coefficient x 1+. 0 Note: The equation has two unequal real roots B2-4ac 0 sector area formula s= 1/2*l*r cone volume formula V= 1/3*S*H cone volume formula V= 1/3*pi*r2h.