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Analytic Geometry Formulas
1. Lines in two dimensions
Line segment
Line forms
Slope - intercept form:
y = mx + b
x = x1 + ( x2 − x1 ) t
Two point form:
y − y1 =
A line segment P1 P2 can be represented in parametric
form by
y2 − y1
( x − x1 )
x2 − x1
Point slope form:
y − y1 = m ( x − x1 )
y = y1 + ( y2 − y1 ) t
0 ≤ t ≤1
Two line segments PP
1 2 and P3 P4 intersect if any only if
the numbers
Intercept form
x y
+ = 1 ( a, b ≠ 0 )
a b
s=
Normal form:
x ⋅ cos σ + y sin σ = p
Parametric form:
x2 − x1
y2 − y1
x3 − x1
x2 − x1
x3 − x4
y3 − y1
y2 − y1
y3 − y4
and
t=
x3 − x1
y3 − y1
x3 − x4
x2 − x1
x3 − x4
y3 − y4
y2 − y1
y3 − y4
satisfy 0 ≤ s ≤ 1 and 0 ≤ t ≤ 1
x = x1 + t cos α
y = y1 + t sin α
Point direction form:
x − x1 y − y1
=
A
B
2. Triangles in two dimensions
Area
The area of the triangle formed by the three lines:
where (A,B) is the direction of the line and P1 ( x1 , y1 ) lies
on the line.
General form:
A ⋅ x + B ⋅ y + C = 0 A ≠ 0 or B ≠ 0
A1 x + B1 y + C1 = 0
A2 x + B2 y + C2 = 0
A3 x + B3 y + C3 = 0
is given by
A1
Distance
The distance from Ax + By + C = 0 to P1 ( x1 , y1 ) is
d=
A ⋅ x1 + B ⋅ y1 + C
2
A +B
K=
2⋅
2
A1
A2
B1
C1
2
A2 B2 C2
A3 B3 C3
B1 A2 B2 A3
⋅
⋅
B2 A3 B3 A1
B3
B1
The area of a triangle whose vertices are P1 ( x1 , y1 ) ,
Concurrent lines
Three lines
A1 x + B1 y + C1 = 0
A2 x + B2 y + C2 = 0
A3 x + B3 y + C3 = 0
are concurrent if and only if:
A1
A2
B1
B2
C1
C2 = 0
A3
B3
C3
P2 ( x2 , y2 ) and P3 ( x3 , y3 ) :
x1
1
K = x2
2
x3
K=
y1 1
y2 1
y3 1
1 x2 − x1
2 x3 − x1
y2 − y1
y3 − y1
.
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Centroid
3. Circle
The centroid of a triangle whose vertices are P1 ( x1 , y1 ) ,
P2 ( x2 , y2 ) and P3 ( x3 , y3 ) :
Equation of a circle
 x + x + x y + y + y3 
( x, y ) =  1 2 3 , 1 2

3
3


In an x-y coordinate system, the circle with centre (a, b)
and radius r is the set of all points (x, y) such that:
( x − a )2 + ( y − b ) 2 = r 2
Circle is centred at the origin
Incenter
x2 + y 2 = r 2
The incenter of a triangle whose vertices are P1 ( x1 , y1 ) ,
Parametric equations
P2 ( x2 , y2 ) and P3 ( x3 , y3 ) :
x = a + r cos t
 ax + bx2 + cx3 ay1 + by2 + cy3 
( x, y ) =  1
,
a + b + c 
 a+b+c
y = b + r sin t
where t is a parametric variable.
In polar coordinates the equation of a circle is:
where a is the length of P2 P3 , b is the length of PP
1 3,
and c is the length of PP
1 2.
r 2 − 2rro cos (θ − ϕ ) + ro2 = a 2
Area
A = r 2π
Circumference
Circumcenter
The
circumcenter
of a triangle whose vertices are
c = π ⋅ d = 2π ⋅ r
P1 ( x1 , y1 ) , P2 ( x2 , y2 ) and P3 ( x3 , y3 ) :
 x12 + y12 y1 1 x1 x12 + y12 1 
 2

2
2
2
 x2 + y2 y2 1 x2 x2 + y2 1 
 x2+y2 y 1 x x2+y2 1
3
3
3
3
3
3

( x, y ) = 
,

x1 y1 1
x1 y1 1 


2 x2 y2 1 
 2 x2 y2 1

x3 y3 1
x3 y3 1 


Theoremes:
(Chord theorem)
The chord theorem states that if two chords, CD and EF,
intersect at G, then:
CD ⋅ DG = EG ⋅ FG
(Tangent-secant theorem)
If a tangent from an external point D meets the circle at
C and a secant from the external point D meets the circle
at G and E respectively, then
2
DC = DG ⋅ DE
Orthocenter
The
orthocenter
of a triangle whose vertices are
P1 ( x1 , y1 ) , P2 ( x2 , y2 ) and P3 ( x3 , y3 ) :
 y1 x2 x3 + y12 1 x12 + y2 y3

2
2
 y2 x3 x1 + y2 1 x2 + y3 y1
 y xx +y2 1 x2+y y
3
1 2
3
3
1 2
( x, y ) = 
,

x1 y1 1
x1 y1

2 x2 y2
 2 x2 y2 1

x3 y3 1
x3 y3

(Secant - secant theorem)
If two secants, DG and DE, also cut the circle at H and F
respectively, then:
DH ⋅ DG = DF ⋅ DE
x1 1 

x2 1 
x3 1 

1 

1 
1 

(Tangent chord property)
The angle between a tangent and chord is equal to the
subtended angle on the opposite side of the chord.
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4. Conic Sections
Eccentricity:
The Parabola
The set of all points in the plane whose distances from a
fixed point, called the focus, and a fixed line, called the
directrix, are always equal.
a 2 − b2
a
Foci:
if a > b => F1 ( − a 2 − b 2 , 0) F2 ( a 2 − b 2 ,0)
The standard formula of a parabola:
y 2 = 2 px
if a < b => F1 (0, − b 2 − a 2 ) F2 (0, b 2 − a 2 )
Parametric equations of the parabola:
x = 2 pt
e=
Area:
2
K = π ⋅ a ⋅b
y = 2 pt
The Hyperbola
Tangent line
2
Tangent line in a point D( x0 , y0 ) of a parabola y = 2 px
y0 y = p ( x + x0 )
Tangent line with a given slope (m)
Tangent lines from a given point
Take a fixed point P ( x0 , y0 ) .The equations of the
tangent lines are
y − y0 = m1 ( x − x0 ) and
y − y0 = m2 ( x − x0 ) where
m1 =
m1 =
2 x0
and
y0 − y0 2 − 2 px0
2 x0
The Ellipse
The set of all points in the plane, the sum of whose
distances from two fixed points, called the foci, is a
constant.
The standard formula of a ellipse
x2 y 2
+
=1
a 2 b2
Parametric equations of the ellipse
x = a cos t
y = b sin t
Tangent line in a point D ( x0 , y0 ) of a ellipse:
x0 x y0 y
+ 2 =1
a2
b
The standard formula of a hyperbola:
x2 y 2
−
=1
a 2 b2
p
y = mx +
2m
y0 + y0 2 − 2 px0
The set of all points in the plane, the difference of whose
distances from two fixed points, called the foci, remains
constant.
Parametric equations of the Hyperbola
a
sin t
b sin t
y=
cos t
x=
Tangent line in a point D ( x0 , y0 ) of a hyperbola:
x0 x y0 y
− 2 =1
a2
b
Foci:
if a > b => F1 ( − a 2 + b 2 , 0) F2 ( a 2 + b 2 , 0)
if a < b => F1 (0, − b 2 + a 2 ) F2 (0, b 2 + a 2 )
Asymptotes:
b
b
x and y = − x
a
a
a
a
if a < b => y = x and y = − x
b
b
if a > b => y =
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5. Planes in three dimensions
Plane forms
Equation of a plane
The equation of a plane through P1(x1,y1,z1) and parallel
to directions (a1,b1,c1) and (a2,b2,c2) has equation
Point direction form:
x − x1 y − y1 z − z1
=
=
a
b
c
where P1(x1,y1,z1) lies in the plane, and the direction
(a,b,c) is normal to the plane.
General form:
Ax + By + Cz + D = 0
where direction (A,B,C) is normal to the plane.
x − x1
y − y1
z − z1
a1
b1
c1
a2
b2
c2
The equation of a plane through P1(x1,y1,z1) and
P2(x2,y2,z2), and parallel to direction (a,b,c), has equation
x − x1
y − y1
z − z1
x2 − x1
y2 − y1
z2 − z1 = 0
a
b
Intercept form:
x y z
+ + =1
a b c
The distance of P1(x1,y1,z1) from the plane Ax + By +
Cz + D = 0 is
d=
Three point form
x − x3
y − y3
z − z3
x1 − x3
y1 − y3
z1 − z3 = 0
x2 − x3
y2 − y3
z2 − z3
Ax1 + By1 + Cz1
A2 + B 2 + C 2
Intersection
The intersection of two planes
A1 x + B1 y + C1 z + D1 = 0,
A2 x + B2 y + C2 z + D2 = 0,
Normal form:
is the line
x cos α + y cos β + z cos γ = p
x − x1 y − y1 z − z1
=
=
,
a
b
c
Parametric form:
x = x1 + a1 s + a2 t
where
y = y1 + b1 s + b2 t
z = z1 + c1 s + c2 t
where the directions (a1,b1,c1) and (a2,b2,c2) are
parallel to the plane.
Angle between two planes:
a=
B1
B2
C1
C2
b=
C1
C2
A1
A2
A1
B1
A2
B2
c=
The angle between two planes:
A1 x + B1 y + C1 z + D1 = 0
b
A2 x + B2 y + C2 z + D2 = 0
x1 =
is
2
1
2
1
A + B +C
2
2
A2 + B2 + C2
The planes are parallel if and only if
y1 =
A1 B1 C1
=
=
A2 B2 C2
The planes are perpendicular if and only if
A1 A2 + B1 B2 + C1C2 = 0
C2
−c
D1
B1
D2
B2
2
D1
D2
A1
A2
2
−c
2
D1
C1
D2
C2
2
a +b +c
a
z1 =
D2
C1
a +b +c
c
2
D1
2
A1 A2 + B1 B2 + C1C2
2
1
c
Distance
this plane passes through the points (a,0,0), (0,b,0), and
(0,0,c).
arccos
=0
D1
D2
B1
B2
2
−b
2
2
D1
A1
D2
A2
a +b +c
2
If a = b = c = 0, then the planes are parallel.