Pedal Triangle
The pedal triangle of \(ABC\) with respect to \(P\)
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 \(ABC\).
Contents
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.
Let \(\Delta ABC\) 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 \(\Delta ABC.\)
The 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.