reason
1
A straight line passing through the midpoint of one waist of the trapezoid and parallel to the bottom will bisect the other waist.
80
reason
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
8 1
Triangle midline theorem
The center line of the triangle is parallel to and equal to the third side.
Half of
82
Trapezoidal midline theorem
The center line of the trapezoid is parallel to the two bottom sides and is equal to the sum of the two bottom sides.
one half
L=
(
a+b
)
÷2s = length× height
83 ( 1)
Basic properties of proportion
if
a:b=c:d，
therefore
AD = BC
if
ad=bc，
therefore
A: B = C: D.
84 (2)
Combined ratio attribute
if
a
/
b=c
/
d，
therefore
(a b)
/
b=(c d)
/
d
85 (3)
Equidistant property
if
a
/
b=c
/
d=…=m
/
n(b+d+…+n≠0)，
therefore
(a+c+…+m)
/
(b+d+…+n)=a
/
b
86
Proportional theorem of parallel line segment
Three parallel lines cut two straight lines, corresponding.
The line segments are proportional.
87
reason
A straight line parallel to one side of a triangle intersects the other two sides (or extension lines of both sides).
, the corresponding line segment is proportional.
88
theorem
If a straight line cuts both sides of a triangle.
(or extension lines on both sides)
The resulting corresponding line segments are proportional,
So this
A straight line is parallel to the third side of the triangle.
Eighty-nine
A straight line parallel to one side of a triangle and intersecting with the other two sides. The three sides of the triangle are the same as the original triangle.
Three sides are proportional.
90
theorem
A straight line parallel to one side and the other two sides of a triangle.
(or extension lines on both sides)
Intersection,
The triangle formed is the same as.
Similarity of primitive triangle
9 1
A Similarity Judgment Theorem of Triangle
1
Two angles are equal and two triangles are similar (
American Statistical Society; usa standards institute
)
92
Two right triangles divided by the height on the hypotenuse are similar to the original triangle.
93
Decision theorem
2
The two sides are proportional and the included angle is equal, and the two triangles are similar (
Scandinavian airlines
)
94
Decision theorem
three
Three sides are proportional and two triangles are similar (
Selective SelectiveServiceSystem
)
95
theorem
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 proportion, and the proportion corresponding to the center line is equal to 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 the similarity ratio.
98
Property theorem
three
The ratio of similar triangles area is equal to the square of similarity ratio.
99
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.
103
The outside of a circle can be regarded as a collection of points whose center is farther than the radius.
104
The same circle or the same circle has the same radius.
105
The distance to the fixed point is equal to the trajectory of the fixed-length point, with the fixed point as the center and half the fixed length.
Diameter circle
106
The locus of points whose distance is equal to the two ends of a known line segment is perpendicular to the line segment.
bisector
107
The locus of a point with equal distance to both sides of a known angle is the bisector of this angle.
108
The locus of a point with equal distance to two parallel lines is parallel to these two parallel lines and at a distance.
A straight line of equality
109
theorem
Three points that are not on the same straight line determine a circle.
1 10
Vertical theorem
The diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite the chord.
1 1 1
reason
1
(1) bisects the diameter (not the diameter) of the chord perpendicular to the chord and bisects the two arcs opposite to 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
reason
2
Two parallel chords of a circle have equal arcs.
1 13
A circle is a central symmetrical figure with the center of the circle as the symmetrical center.
1 14
theorem
In the same circle or in the same circle, the arcs with equal central angles are equal, and so are the chords.
Equal, the chord center distance of the opposite chord is equal.
1 15
reason
In the same circle or equal 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.
1 16
theorem
An arc subtends a circumferential angle equal to half the central angle it subtends.
1 17
reason
1
The circumferential angles of the same arc or equal arc are equal; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.
1 18
reason
2
The circumference angle (or diameter) of a semicircle is a right angle;
90
Yuanzhoujiao station
The chord on the right is the diameter.
1 19
reason
three
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 it.
Internal diagonal of
12 1
① straight line
L
Heba
O
stride
d
r
122
Tangent judgment theorem
The straight line passing through the outer end of the radius and perpendicular to the radius is the tangent of the circle.
123
Property theorem of tangent line
The tangent of a circle is perpendicular to the radius passing through the tangent point.
124
reason
1
A straight line passing through the center of the circle and perpendicular to the tangent must pass through the tangent point.
125
reason
2
A straight line that passes through the tangent and is perpendicular to the tangent must pass through the center of the circle.
126
Tangent length theorem
Two tangents drawn from a point outside the circle are equal in length,
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
Alternating line segment theorem
The chord tangent angle is equal to the circumferential angle of the arc pair it clamps.
129
reason
If the arc enclosed by two chord angles is equal, then the two chord angles are also equal.
130
the intersection string theorem
The length of two intersecting chords in a circle divided by the product of the intersection points.
(to) equal to ...
13 1
reason
If a chord intersects a diameter perpendicularly, half of the chord is formed by its minor diameter.
Proportional median of two line segments
132
Tangent secant theorem
The tangent and secant of the circle are drawn from a point outside the circle, and the length of the tangent is the tangent from that point to the tangent.
The proportional average of the lengths of two straight lines at the intersection of a straight line and a circle.
133
reason
The product of two secant lines that form a circle from a point outside the circle to the intersection of each secant line and the circle.
wait for
134
If two circles are tangent, then the tangent point must be on the line.
135
① The two circles are separated from each other.
d
>
R+r
(2) circumscribe two circles.
d=R+r
③ Two circles intersect.
R-r
r)
⑤ Two circles contain.
d
r)
136
theorem
The line of intersection with two circles bisects the common chord of the two circles vertically.
137
theorem
Divide a circle into
n(n≥3):
(1) The polygon obtained by connecting the points in turn is the inscribed circle of this circle.
n
Edge shape
⑵ The tangent of the circle passing through each point, and the polygon whose vertex is the intersection of adjacent tangents is the circumscribed circle.
n
Edge shape
138
theorem
Any regular polygon has a circumscribed circle and an inscribed circle, which are concentric circles.
139
straight
n
Each inner angle of a polygon is equal to (
n-2
)
× 180
/
n
140
theorem
straight
n
The radius and apogee of a polygon are positive.
n
edge segmentation
2n
Congruent right triangle
14 1
straight
n
Area of polygon
Sn=pnrn
/
2 pence
Express affirmation
n
Perimeter of polygon
142
Regular triangle area
√3a
/
4 a
Represents the side length
143
If there is a vertex
k
Ge Zheng
n
The angles of a polygon, because the sum of these angles should be
360
, therefore
k×(n-2) 180
/
n=360
Become (
n-2
)
(k-2)=4
144
Arc length calculation formula:
L=n
Rise to a height
rare
/
180
145
Sector area formula:
S
department
=n
Rise to a height
R^2
/
360=LR
/
2
146
Internal common tangent length
= d-(R-r)
External 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
| One-
b|≥|a|
-|b| -
|a|≤a≤|a|
Solution of quadratic equation in one variable
-
b+√(b2
-4ac)/2a -b-
√(b2
-4ac)/2a
Relationship between root and coefficient
x 1+X2 =-b/a x 1 * X2 = c/a
Note: Vieta theorem.
discriminant
b2-4ac=0
Note: The equation has two equal real roots.
b2-4ac >0
Note: The equation has two unequal real roots.
B2-4ac & lt； 0
Note: The equation has no real root, but has a plurality of yokes.
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
-
√(( 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
Before some series,
n
peaceful and auspicious
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+2
3+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:
In ...
rare
Represents the radius of the circumscribed circle of a triangle.
cosine theorem
B2 = a2+C2-2 acco b
Note: Angle
B
Bian Shi
a
Bian He
c
Angle of
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 & gt； 0
Parabolic standard equation
y2=2px y2=-2px x2=2py x2=-2py
Transverse area of right prism
S=c*h
Oblique prism side area
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
Arc length formula
l=a*r a
Is the radian number of the central angle.
r & gt0
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 area of the straight line,
L is the length of the side.
Cylinder volume formula
V=s*h
cylinder
V=pi*r2h
Sin30: half
Sin45: Two thirds of the root.
Sin60: two-thirds of the root number 3
Cos30: two-thirds root three
Cos45: two-thirds root number two
half
Tan30: two thirds of the root
Cos45: one
Tan60: Genshan
Geometric series:
If q = 1
rule
S=n*a 1
if
q≠ 1
Overturning process:
s=a 1+a 1*q+a 1*q^2+……+a 1*q^(n- 1)
Multiply both sides of the equation by q at the same time
S*q=a 1*q+a 1*
q^2+a 1*q^3+……+a 1*q^
1 type -2 has
s=a 1*( 1-q^n)/( 1-q)
arithmetic series
Deduction process:
s = a 1+(a 1+d)+(a 1+2d)+……(a 1+(n
- 1)*d)
Write this formula backwards.
s =(a 1+(n- 1)* d)+(a 1+(n-2)* d)+(a 1+(n-
3)*d)+……+a 1
The above two formulas add up to s = (2a1+(n-1) d) * n/2 = n * a1+n * (n-1) * d/2.