ANALYTIC GEOMETRY FORMULAS
Analytic Geometry is a branch of algebra, an invention of Descartes and Fermat. It deals with modelling some geometrical objects, such as lines, points, curves, and so on. The study of analytic geometry is important as it gives the knowledge for the next level of mathematics.
LINES IN TWO DIMENSIONS
Slope-intercept form
Two point form
Point slope form
Intercept form
Normal form
Concurrent lines Three lines
are concurrent if and only if:
Parametric form
Point direction form
where (A, B) is the direction of the line and P1 (X1,Y1) lies on the line.
General form
Distance
Line segment
satisfy 0 ≤ s ≤ 1 and 0 ≤ t ≤ 1
TRIANGLES IN TWO DIMENSIONS
Area
The area of the triangle formed by the three lines: 
is given by 
The area of a triangle whose vertices are 
Centroid
The centroid o f a triangle whose vertices are
Incenter
The vertices of a triangle whose incenter are
Circumcenter
The circumcenter of a triangle whose vertices are
Orthocenter
The orthocenter of a triangle whose vertices are
CIRCLE
Equation of a circle
The circle with the center (a, b) and radius r in a x-y coordinate system in the set of all points ( x, y) such that:
Circle is the centered at the region
Parametric equations
where is the parametric variable. In polar coordinates the equation of a circle is:
Area
Circumference
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 the tangent from an external point D intersects the circle at C and a secant from the external point D intersects the circle at G and E, then:
DC² = DG • DE
The secant-secant theorem: states that if two secants, DG & DE, cut the circle at H and F, respectively, then
DH •DG = DF • DE
Property of the tangent chord: The angle formed by a tangent and a chords is the equal to the subtended angle on the chord’s opposite side.
CONIC SECTION
The Parabola
The set all points in the plane whose distances from a fixed point, referred to as the focus, and a fixed line, known as the directrix, are always equal.
A parabola’s standard formula is:
The parabola’s parametric equations:
Tangent line
Tangent line with a given slope (m)
Tangent lines from a given point
Eccentricity:
Foci:
Area:
K= π • a • b
The Hyperbola
All points in the plane whose distances from the two fixed points, known as foci, stay constant
The hyperbola’s standard formula
The hyperbola’s parametric equation
Foci:
Asymptotes:
PLANES IN THREE DIMENSIONS
Plane forms
Point direction form:
where ( x1, y1, z1) lies in the plane, and the direction (a, b, c) is normal plane.
General form:
Ax + By + Cz + D = 0
where direction ( A, B,C) is the normal to the plane.
Intercept form:
this plane passes through the points ( a, 0,0), (0,b,0), and (0,0,c)
Three point form:
Normal form:
Parametric form:
Angle between two planes:
The angle between two planes
The planes are parallel id and only if
The planes are perpendicular if and only if
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)
The equation of a plane through P1 (X1, Y1, Z1) and P2 (a2, b2, c2) and parallel to the direction (a, b, c), has equation.
Distance
The distance of P1 (x1, y1, z1) from the plane Ax + By + Cz + D= 0 is
Intersection
The intersection of two planes
A1y + B1y + C1z +D1 = 0,
A2x + B2y + C2z + D2 =0
is the line
