Equation of a circle

In an x−y coordinate system, the circle with center (a,b) and radius r is the set of all points (x,y) such that:

(x−a)2+(y−b)2=r2

Circle centered at the origin:

x2+y2=r2

Parametric equations

xy=a+rcost=b+rsint

where t is a parametric variable.

In polar coordinates the equation of a circle is:

r2−2⋅r⋅r0⋅cos(Θ−ϕ)+r20=a2

Area of a circle

A=r2π

Circumference of a circle

C=π⋅d=2⋅π⋅r

Theorems:

(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

DC2=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.