Theorem 3 Two figures are symmetrical about a straight line. If their corresponding line segments or extension lines intersect, the intersection point is on the axis of symmetry. Inverse Theorem 45 If the 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. Pythagorean Theorem 46 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. The sum of the quadrilateral internal angles of Theorem 48 is equal to 360 49, the sum of the polygon internal angles of Theorem 360 50 is equal to (n-2) × 180 5 1, and the sum of the external angles of any polygon is equal to the diagonal phase of the parallelogram of Theorem 1. Equality 53 parallelogram property theorem 2 parallelogram with equal sides 54 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 groups of parallelograms with equal sides are parallelograms. Parallelogram Decision Theorem of Side 58 3 A quadrilateral whose diagonal is bisected is a parallelogram 59. Parallelogram Decision Theorem 4 A group of parallelograms whose opposite sides are parallel is a parallelogram 60. Rectangular property theorem 1 Rectangular property theorem 6 1 Rectangular diagonal is equal to 62. Rectangular decision theorem 1 Quadrilateral whose three corners are right angles. It is a rectangle. Rectangular decision theorem ii. The parallelogram with equal diagonal lines is a rectangle 64. The 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 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 A straight line passing through the midpoint of one waist of a trapezoid and parallel to the bottom will bisect the other waist. Inference 2 A straight line passing through the midpoint of one side of a triangle and parallel to the other side will split the first side in two.
The midline theorem of a trilateral triangle 8 1 is parallel to the third side and equal to half of the third side. The midline theorem of trapezoid is parallel to the two bottoms and equal to half of the sum of the two bottoms. Basic properties of L=(a+b)÷2 S=L×h 83 (1) If a:b=c:d, then a:b=c:d 84 (2) If a/b = c/d, then (a b)/b = (. 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 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, with the fixed-length point as the center, the locus of a circle with a fixed length of half diameter 106, the locus of a point with the same distance from the middle vertical line of the line segment 107 to the points with the same distance from both sides of the known angle, and the locus of a point with the same distance from the bisector of the angle 108 to two parallel lines are parallel.
109 The theorem of starting from an equal straight line determines the circles at three points that are not on the same straight line. 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. 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.
Theorem 137 divides the circle into n (n ≥ 3): (1) The polygon obtained by connecting the points in turn is a regular n polygon inscribed in the circle; (2) The tangent of a circle passing through a point, and the polygon whose vertex is the intersection of adjacent tangents is the circumscribed regular N polygon of the circle; (2) Theorem 138: Any regular polygon has an inscribed circle and an inscribed 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. Practical tools: common mathematical formulas, 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) triangle inequality. -b ≤ a ≤ b | a-b |≥| a |-b |-a |≤ a | the solution of a quadratic equation with one variable -b+√(B2-4ac)/2a-b-√(B2-4ac)/2a root and the coefficient x 1+. 0 Note: This equation has two unequal real roots B2-4ac.
Cos2a = cos2a-sin2a = 2cos2a-1=1-2sin2a half-angle formula sin (a/2) = √ ((1-COSA)/2) sin (a/2) =-√ ((/kloc-) tan(A/2)=√(( 1-COSA)/((65438-SIN(A-B)-2 sinA sinB = 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)/COSA COSB Tana-Tanb = SIN (A-B)/COSA COSB CTGA+CTG BSIN (A+B)/SINA SINB-CTGA+CTG BSIN (A+B)/SINA SINB before N terms and 6N2/KLOC. Kloc-0/1+13+15+…+(2n-1) = N2 2+4+6+8+10+12+/. 0 parabola standard equation y2 = 2px2 =-2px2 =-2px lateral area S=c*h Oblique prism lateral area S=c'*h Regular pyramid lateral area S= 1/2c*h' Regular pyramid lateral area S= 1/2(c+c')h' Cone lateral area. L=pi(R+r)l Surface area of a sphere S=4pi*r2 Cylindrical lateral area S=c*h=2pi*h Conical 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 =/.
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
I am tired! The landlord should add more points! ! ! !