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:
\( (x-a)^2 + (y-b)^2 =r^2 \)
Circle centered at the origin:
\( x^2 + y^2 = r^2 \)
Parametric equations
\( \begin{aligned} x &= a + r\,\cos t \\ y&= b + r\,\sin t \end{aligned} \)
where t is a parametric variable.
In polar coordinates the equation of a circle is:
\( r^2 – 2\cdot r \cdot r_0\cdot cos(\Theta – \phi ) + r_0^2 = a^2 \)
Area of a circle
\( A = r^2\pi \)
Circumference of a circle
\( C = \pi \cdot d = 2\cdot \pi \cdot r \)
Theorems:
Chord theorem
The chord theorem states that if two chords, CD and EF, intersect at G, then:
\( CD \cdot DG = EG \cdot FG \)
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
\( DC^2 = DG \cdot DE \)
Secant – secant theorem
If two secants, DG and DE, also cut the circle at H and F respectively, then:
\( DH \cdot DG = DF \cdot DE \)
Tangent chord property
The angle between a tangent and chord is equal to the subtended angle on the opposite side of the chord.