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

$$ (xa)^2 + (yb)^2 =r^2 $$ 
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:

$$ r^2  2\cdot r \cdot r_0\cdot cos(\Theta  \phi ) + r_0^2 = a^2 $$ 

$$ A = r^2\pi $$ 

$$ C = \pi \cdot d = 2\cdot \pi \cdot r $$ 
(Chord theorem) The chord theorem states that if two chords, $CD$ and $EF$, intersect at $G$, then:

$$ CD \cdot DG = EG \cdot FG $$ 
(Tangentsecant 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.
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