The line segment between two points is the shortest.

The complementary angles of the same angle or equal angle are equal.

The complementary angles of the same angle or the same angle are equal.

One and only one straight line is perpendicular to the known straight line.

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 Parallel axiom passes through a point outside a straight line, and there is only one straight line parallel to this straight line.

If both lines are parallel to the third line, the two lines are also parallel to each other.

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

The internal dislocation angles of 10 are equal, and the two straight lines are parallel.

1 1 are complementary and 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.

Theorem 15 The sum of two sides of a triangle is greater than the third side.

16 infers that the difference between two sides of a triangle is smaller than the third side.

The sum of the internal angles of 17 triangle is equal to 180.

18 infers 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 non-adjacent inner angles.

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

2 1 congruent triangles has equal sides and angles.

Axiom of Angular (SAS) has two triangles with equal angles.

The Axiom of 23 Angles (ASA) has the congruence of two triangles, which have two angles and their sides correspond to each other.

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

The axiom of 25 sides (SSS) has two triangles with equal sides.

Axiom of hypotenuse and right angle (HL) Two right angle triangles with hypotenuse and right angle are congruent.

Theorem 1 The distance between a point on the bisector of an angle and both sides of the angle is equal.

Theorem 2 is a point with equal distance on both sides of an angle, which is on the bisector of this angle.

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

The nature theorem of isosceles triangle 30 The two base angles of isosceles triangle are equal (that is, equilateral and equiangular).

3 1 Inference 1 The bisector of the vertices of an isosceles triangle bisects the base and is perpendicular to the base.

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.

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

34 Judgment Theorem of an isosceles triangle If a triangle has two equal angles, then the opposite sides of the two angles are also equal (equal angles and equal sides).

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.

In a right triangle, if an acute angle is equal to 30, the right side it faces is equal to half of the hypotenuse.

The center line of the hypotenuse of a right triangle is equal to half of the hypotenuse.

Theorem 39 The distance between the point on the vertical line of a line segment and the two endpoints of the line segment is equal.

The inverse theorem and the point where the two endpoints of a line segment are equidistant are on the middle vertical line of this line segment.

The perpendicular bisector of the 4 1 line segment can be regarded as the set of all points with equal distance from both ends of the line segment.

Theorem 42 1 Two graphs symmetric about a line are conformal.

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

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.

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 angles A and B of a right triangle is equal to the square of the hypotenuse C, that is, A 2+B 2 = C 2.

47 Inverse Theorem of Pythagorean Theorem If the three sides of a triangle A, B and C are related in length A 2+B 2 = C 2, then the triangle is a right triangle.

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

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

The theorem of the sum of internal angles of 50 polygons is that the sum of internal angles of n polygons 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 equality

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

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 decision theorem 1 Two groups of parallelograms with equal diagonals are parallelograms.

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

58 parallelogram decision theorem 3 A quadrilateral whose diagonal is bisected is a parallelogram.

59 parallelogram decision theorem 4 A group of parallelograms with equal opposite sides are parallelograms.

60 Rectangle Property Theorem 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 Parallelograms with equal diagonals are rectangles

64 diamond property theorem 1 all four sides of the 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 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 whose diagonals are perpendicular to each other are diamonds.

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

Theorem of 70 Square Properties 2 Two diagonal lines of a square are equal and bisected vertically, and each diagonal line bisects a set of diagonal lines.

Theorem 7 1 1 is congruent with respect to two centrosymmetric graphs.

Theorem 2 About two graphs with central symmetry, the connecting lines of symmetric points both pass through the symmetric center and are equally divided by the symmetric center.

Inverse Theorem If a straight line connecting the corresponding points of two graphs passes through a point and is bisected by the point, then the two graphs are symmetrical about the point.

The property theorem of isosceles trapezoid is that two angles of isosceles trapezoid on the same base are equal.

The two diagonals of an isosceles trapezoid are equal.

76 isosceles trapezoid decision theorem A trapezoid with two equal angles on the same base is an isosceles trapezoid.

A trapezoid with equal diagonal lines is an isosceles trapezoid.

Theorem of Equal Segment of Parallel Lines If a group of parallel lines have equal segments on a straight line, the segments on other straight lines also have equal segments.

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 bisect the third side.

The median line theorem of 8 1 triangle The median line of a triangle is parallel to the third side and equal to half of it.

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

83 (1) Basic Properties If a:b=c:d, then ad=bc If ad=bc, then A: B = C: D.

84 (2) Combinatorial Properties If A/B = C/D, then (A B)/B = (C D)/D.

85 (3) Isometric property If A/B = C/D = … = M/N (B+D+…+N ≠ 0), then (A+C+…+M)/(B+D+…+N) = A/B.

86 parallel lines are divided into segments and the theorem of proportionality. Three parallel lines cut two straight lines, and the corresponding segments are proportional.

It is inferred that the line parallel to one side of the triangle cuts the other two sides (or the extension lines of both sides), and the corresponding line segments are 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 this straight line is parallel to the third side of the triangle.

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.

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 triangles's decision theorem 1 Two angles are equal and two triangles are similar (ASA)

Two right triangles divided by the height on the hypotenuse are similar to the original triangle.

Decision Theorem 2: Two sides are proportional and the included angle is equal, and two triangles are similar (SAS).

Decision Theorem 3 Three sides are proportional and 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.

96 Property Theorem 1 similar triangles corresponds to the height ratio, the ratio corresponding to the median line and the ratio corresponding to the angular bisector are all equal to the similarity ratio.

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

98 Property Theorem 3 The ratio of similar triangles area is equal to the square of similarity ratio.

The sine of any acute angle is equal to the cosine of the remaining angles, and the cosine of any acute angle is equal to the sine 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.

The outer circle of 103 circle can be regarded as a collection of points whose center distance is greater than the radius.

104 The radius of the same circle or equal circle is the same.

The distance from 105 to the fixed point is equal to the trajectory of a fixed-length point, which is a circle with the fixed point as the center and the fixed length as the radius.

106 and the locus of the point with the same distance between the two endpoints of the known line segment is the middle vertical line of the line segment.

The locus from 107 to a point with equal distance on both sides of a known angle is the bisector of this angle.

The trajectory from 108 to the equidistant point of two parallel lines is a straight line parallel and equidistant to these two parallel lines.

Theorem 109 Three points that are not on the same straight line determine a circle.

1 10 vertical diameter theorem divides the chord perpendicular to its diameter into two parts, and divides the two arcs opposite to the chord into two parts.

1 1 1 inference 1 ① bisect the diameter of the chord (not the diameter) perpendicular to the chord and bisect the two arcs opposite 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 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.

Theorem 1 14 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 equal circle, if one set of quantities in two central angles, two arcs, two chords or the chord-center distance between two chords is equal, the corresponding other set of quantities is also equal.

Theorem 1 16 The angle of an arc is equal to half its central angle.

1 17 Inference 1 The circumferential angles of the same arc or the same arc are equal; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.

1 18 Inference 2 The circumferential angle (or diameter) of a 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 Theorem The diagonals of the inscribed quadrilateral of a circle are complementary, and any external angle is equal to its internal angle.

12 1① the intersection of the straight line l and ⊙O is d < r.

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

③ lines l and ⊙O are separated by d > r.

122 tangent theorem The straight line passing through the outer end of the radius and perpendicular to the radius is the tangent of the circle.

123 The property theorem of tangent line The tangent line 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.

The tangent length theorem 126 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 a circle's circumscribed quadrilateral is equal.

128 Chord Angle Theorem The chord angle is equal to the circumferential angle of the arc pair it clamps.

129 Inference: If the arc enclosed by two chord tangent angles is equal, then the two chord tangent angles are also equal.

130 intersection chord theorem The length of two intersecting chords in a circle divided by the product of the intersection point is equal.

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132 tangent theorem leads to the tangent and secant of a circle from a point outside the circle, and the tangent length is the middle term in the length ratio of the two lines at the intersection of this point and secant.

133 It is inferred that two secant lines of the circle are drawn from a point outside the circle, and the product of the lengths of the two lines from that point 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.

135① perimeter of two circles D > R+R ② perimeter of two circles d = r+r.

③ the intersection of two circles r-r < d < r+r (r > r).

④ inscribed circle D = R-R (R > R) ⑤ two circles contain D < R-R (R > R).

Theorem 136 The intersection of two circles bisects the common chord of two circles vertically.

Theorem 137 divides a 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 138 Any regular polygon has a circumscribed circle and an inscribed circle, which are concentric circles.

139 every inner angle of a regular n-polygon is equal to (n-2) ×180/n.

140 Theorem Radius and apothem Divides a regular N-polygon into 2n congruent right triangles.

14 1 the area of the regular n polygon Sn = PNRN/2 P represents the perimeter of the regular n polygon.

142 The area of a regular triangle √ 3a/4a indicates the side length.

143 if there are k positive n corners around a vertex, since the sum of these angles should be 360, then K× (n-2) 180/n = 360 becomes (n-2)(k-2)=4.

The formula for calculating the arc length of 144 is L = nπ r/ 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)

Multiplication and factorization

a2-B2 =(a+b)(a-b)a3+B3 =(a+b)(a2-a b+B2)a3-B3 =(a-b)(a2+a b+B2)

Triangle inequality

a+b |≤| a |+| b | | a-b |≤| a |+| b | | a |≤b & lt； = & gt-b≤a≤b |a-b|≥|a|-|b| -|a|≤a≤|a

Solution of quadratic equation in one variable

-b+√(b2-4ac)/2a -b-√(b2-4ac)/2a

The relationship between roots and coefficients (Vieta theorem)

x 1+X2 =-b/a x 1 * X2 = c/a

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+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)

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)/cosa cosb tanA-tanB = sin(A-B)/cosa cosb

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 law

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.

geometry

the standard equation of the circle

(x-a)2+(y-b)2=r2 Note: (a, b) is the central coordinate.

Circular general equation

X2+y2+Dx+Ey+F=0 Note: D2+E2-4f > 0

Parabolic standard equation

y2=2px y2=-2px x2=2py x2=-2py

Transverse area of right prism

S=c*h side area of oblique prism S=c'*h

Side area of regular pyramid

S= 1/2c*h '

Transverse area of regular prism

S= 1/2(c+c')h '

Yuantai lateral area

s = 1/2(c+c’)l = pi(R+R)l

Surface area of ball

S=4pi*r2

Cylindrical side area

S=c*h=2pi*h

Cone lateral area

S= 1/2*c*l=pi*r*l

Diamond area

S= bottom * height S= 1/2* diagonal product

Arc length formula

L=a*r a is the radian number of the central angle r >; 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

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