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