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 $ABC$.

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.

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.

• Incircles and Excircles
• Circumcircle of Triangle
• Orthic Triangle
• Simson Line Theorem