Math formulas: Circle

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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.