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 the corresponding points of two graphs pass through a certain point and are connected by it.
If the point is split in two, 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 bisecting line segments by parallel lines If a group of parallel lines are tangent to a straight line.
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 be equally divided.
Trilaterality
The median line theorem of 8 1 triangle The median line of a triangle is parallel to and equal to the third side.
Half of
The trapezoid midline theorem is parallel to the two bottoms and equals the sum of the two bottoms.
Half l = (a+b) ÷ 2s = l× h。
Basic properties of ratio 83 (1) 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 proportional theorems. Three parallel lines cut two straight lines and get the corresponding results.
The line 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 of a right triangle and one right-angled side and another right-angled side
The hypotenuse of an angle is proportional to a right-angled side, so two right-angled triangles are similar.
96 Property Theorem 1 similar triangles has a high ratio, and the ratio corresponding to the center line is flat with the corresponding angle.
The ratio of dividing lines is 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 value of any acute angle is equal to the cosine value of other angles, the cosine value of any acute angle, etc.
Sine value of other angles
100 The tangent of any acute angle is equal to the cotangent of other angles, the cotangent of any acute angle, etc.
Tangent value of its complementary angle
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.
105 The distance from the fixed point is equal to the trajectory of the fixed point, with the fixed point as the center, and the fixed length is half.
Diameter circle
106 and it is known that the locus of the point with the same distance between the two endpoints of the line segment is perpendicular to the line segment.
bisector
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 point with the same distance from two parallel lines is parallel to these two parallel lines with a distance of.
A straight line of equality
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, arcs with equal central angles are equal, and chords with equal central angles are equal.
Equal, the chord center distance of the opposite chord is equal.
1 15 inference in the same circle or in the same circle, if two central angles, two arcs, two chords or two.
If one set of quantities in the chord-to-chord distance is equal, then the other sets of quantities corresponding to it are 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; 90 degree circle angle
The chord on the right is the 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 inscribed quadrilateral of a circle is diagonally complementary, and any external angle is equal to it.
Internal diagonal of
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.
126 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 between 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 product of two intersecting chords in a circle divided by the intersection point.
(to) equal to ...
13 1 Inference: If the chord intersects the diameter vertically, then half of the chord is formed by dividing it by the diameter.
Proportional median of two line segments
132 tangent theorem leads to the tangent and secant of a circle from a point outside the circle, and the tangent length is the point to be cut.
The proportional average of the lengths of two straight lines at the intersection of a straight line and a circle.
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, then the sum of these angles should be
360, so k× (n-2) 180/n = 360 is changed to (n-2)(k-2)=4.
The formula for calculating the arc length of 144 is 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)
There are still some, please help to supplement them. )
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))
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+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 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=2px y2=-2px x2=2py x2=-2py
Lateral area of a straight prism S=c*h lateral area of an 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 of a cone 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*r a 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