# Pedal Triangle

The **pedal triangle** of a triangle $ABC$ and point $P$ is the triangle whose vertices are the projections of a $P$ to the sides of the triangle. For instance, when $P$ is the orthocenter of the triangle, the pedal triangle is the orthic triangle; when the term is used without further qualification, this is often what is meant by "the" pedal triangle of $ABC$.

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## Properties

Any pedal triangle $DEF$ satisfies

$AD^2 + BE^2 + CF^2 = BD^2 + CE^2 + AF^2.$

If $P$ lies on the circumcircle of the triangle, the pedal triangle becomes the Simson line with respect to $P$. This is useful both for showing that $P$ lies on the circumcircle, and for showing three points are collinear.

$\Delta ABC$ be a triangle with vertices $A=(0,0), B=(6,0), C=\big(3,3\sqrt{3}\big),$ and $\Delta PQR$ the pedal triangle of $\Delta ABC.$

LetThe sum of the circumradius of $\Delta ABC$ and inradius of $\Delta PQR$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a,b,$ and $c,$ where $a,c$ are coprime and $b$ is square-free.

Find $a\times b\times c$.

Further,

The area of the pedal triangle is given by

$\left|\frac{R^2-OP^2}{4R^2}\right|,$

where $R$ is the radius of the circumcircle, and $O$ is the circumcenter.