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

(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 $$ DC^2 = DG \cdot DE $$ Chord theorem

(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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