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Analytic Geometry Formulas
1. Lines in two dimensions
Line forms Slope - intercept form: y = mx + b Two point form:
1 2 1 (^1 )
2 1
y y y^ y x x x x
Point slope form:
y − y 1 = m x ( − x 1 )
Intercept form
a^ x^^ +^ by^ =^1 (^ a b ,^ ≠^0 )
Normal form:
x ⋅ cos σ + y sinσ= p
Parametric form: 1 1
cos sin
x x t y y t
Point direction form: x x 1 (^) y y 1 A B
where (A,B) is the direction of the line and P x 1 ( 1 (^) , y 1 )lies on the line. General form: A x ⋅ + B y ⋅ + C = 0 A ≠ 0 or B ≠ 0
Distance
The distance from Ax + By + C = 0 to P x 1 ( 1 (^) , y 1 )is 1 1 2 2 d A x^ B y^ C A B
=^ ⋅^ +^ ⋅^ +
Concurrent lines Three lines 1 1 1 2 2 2 3 3 3
A x B y C A x B y C A x B y C
are concurrent if and only if: 1 1 1 2 2 2 3 3 3
A B C
A B C
A B C
Line segment A line segment P P 1 2 can be represented in parametric form by
1 2 1 1 2 1 0 1
x x x x t y y y y t t
Two line segments (^) PP 1 2 and (^) P P 3 4 intersect if any only if the numbers 2 1 2 1 3 1 3 1 3 1 3 1 3 4 3 4 2 1 2 1 2 1 2 1 3 4 3 4 3 4 3 4
x x y y x x y y s x^ x^ y^ y^ and t x^ x^ y^ y x x y y x x y y x x y y x x y y
= −^ −^ = −^ −
satisfy 0 ≤ s ≤ 1 and 0 ≤ t ≤ 1
2. Triangles in two dimensions
Area The area of the triangle formed by the three lines: 1 1 1 2 2 2 3 3 3
A x B y C A x B y C A x B y C
is given by 2 1 1 1 2 2 2 3 3 3 1 1 2 2 3 3 2 2 3 3 1 1
A B C
A B C
A B C
K A B A B A B
A B A B A B
The area of a triangle whose vertices are P x 1 ( 1 (^) , y 1 ), P 2 (^) ( x 2 (^) , y 2 )and P 3 (^) ( x 3 (^) , y 3 ): 1 1 2 2 3 3
x y K x y x y
2 1 2 1 3 1 3 1
K x^ x^ y^ y x x y y
= −^ −
Centroid
The centroid of a triangle whose vertices are P x 1 ( 1 (^) , y 1 ), P 2 (^) ( x 2 (^) , y 2 )and P 3 (^) ( x 3 (^) , y 3 ):
( , ) 1 2 3 ,^1 2 3 3
x y = ^^ x^ +^ x^ +^ x^ y^ +^ y^ + y
Incenter
The incenter of a triangle whose vertices are P x 1 ( 1 (^) , y 1 ), P 2 (^) ( x 2 (^) , y 2 )and P 3 (^) ( x 3 (^) , y 3 ):
( , x y )^ ax^1^^ bx^2^^ cx^3^^ , ay^1^^ by^2^^ cy^3 a b c a b c
= ^ +^ +^ +^ +
where a is the length of P P 2 3 , b is the length of PP 1 3 , and c is the length of (^) PP 1 2.
Circumcenter The circumcenter of a triangle whose vertices are P x 1 ( 1 (^) , y 1 ), P 2 (^) ( x 2 (^) , y 2 )and P 3 (^) ( x 3 (^) , y 3 ):
12 12 1 1 12 12 22 2 2 2 2 2 2 22 32 32 3 3 3 2 32 1 1 1 1 2 2 2 2 3 3 3 3
x y y x x y x y y x x y x y y x x y x y (^) x y x y x y x y x y x y
+^ +
= ^
Orthocenter The orthocenter of a triangle whose vertices are P x 1 ( 1 (^) , y 1 ), P 2 (^) ( x 2 (^) , y 2 )and P 3 (^) ( x 3 (^) , y 3 ): 1 2 3 12 12 2 3 1 2 3 1 22 2 2 3 1 2 3 1 2 32 32 1 2 3 1 1 1 1 2 2 2 2 3 3 3 3
y x x y x y y x y x x y x y y x y x x y x y y x x y (^) x y x y x y x y x y x y
+^ +
= ^
3. Circle
Equation of a circle 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 x^2^ + y^2 = r^2 Parametric equations cos sin
x a r t y b r t
where t is a parametric variable. In polar coordinates the equation of a circle is: r^2 − 2 rro cos ( θ − ϕ)+ ro^2 = a^2 Area
A = r^2 π
Circumference
c = π ⋅ d = 2 π⋅ r
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 DC^2 = DG ⋅ DE (Secant - secant theorem) If two secants, DG and DE, also cut the circle at H and F respectively, then: DH ⋅ DG = DF ⋅ DE (Tangent chord property) The angle between a tangent and chord is equal to the subtended angle on the opposite side of the chord.
5. Planes in three dimensions
Plane forms
Point direction form: x x 1 (^) y y 1 (^) z z 1 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. Intercept form: x y z 1 a b c
this plane passes through the points (a,0,0), (0,b,0), and (0,0,c). Three point form 3 3 3 1 3 1 3 1 3 2 3 2 3 2 3
x x y y z z x x y y z z x x y y z z
Normal form:
x cos α+ y cos β + z cosγ= p
Parametric form: 1 1 2 1 1 2 1 1 2
x x a s a t y y b s b t z z c s c t
where the directions (a1,b1,c1) and (a2,b2,c2) are parallel to the plane.
Angle between two planes: The angle between two planes: 1 1 1 1 2 2 2 2
A x B y C z D A x B y C z D
is 1 2 1 2 1 2 arccos (^12 12 12 22 2 2 )
A A B B C C
A B C A B C
The planes are parallel if and only if 1 1 1 2 2 2
A B C
A B C
The planes are perpendicular if and only if A A 1 2 (^) + B B 1 2 (^) + C C 1 2 = 0
Equation of a plane The equation of a plane through P 1 (x 1 ,y 1 ,z 1 ) and parallel to directions (a 1 ,b 1 ,c 1 ) and (a 2 ,b 2 ,c 2 ) has equation 1 1 1 1 1 1 2 2 2
x x y y z z a b c a b c
The equation of a plane through P 1 (x 1 ,y 1 ,z 1 ) and P 2 (x 2 ,y 2 ,z 2 ), and parallel to direction (a,b,c), has equation 1 1 1 2 1 2 1 2 1 0
x x y y z z x x y y z z a b c
Distance The distance of P1(x1,y1,z1) from the plane Ax + By + Cz + D = 0 is 1 1 1 2 2 2 d^ Ax^ By^ Cz A B C
=^ +^ +
Intersection The intersection of two planes 1 1 1 1 2 2 2 2
A x B y C z D A x B y C z D
is the line x x 1 (^) y y 1 (^) z z (^1) , a b c
where 1 1 2 2
a B^ C B C
1 1 2 2
C A
b = C A
1 1 2 2
c A^ B A B
1 1 1 1 2 2 2 2 (^1 2 2 )
D C D B
b (^) D C cD B x a b c
1 1 1 1 1 2 2 2 2 2 2 2
c D^ A^ cD^ C y D^ A^ D^ C a b c
1 1 1 1 1 2 2 2 2 2 2 2
a D^ B^ bD^ A z D^ B^ D^ A a b c
If a = b = c = 0, then the planes are parallel.
