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Math Formulas: Circle

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.