# Orthic Triangle

The **orthic triangle** of a triangle $ABC$ is the triangle whose vertices are the feet of the altitudes from $A, B,$ and $C$ to the opposite sides. It is also the pedal triangle of the orthocenter.

#### Contents

## Properties

Denote the feet of the altitudes by $A', B',$ and $C',$ respectively. Then

$\angle AA'B' = \angle AA'C',\quad \angle BB'C' = \angle BB'A',\quad \angle CC'A' = \angle CC'B',$

which can be summarized as follows:

The incenter of the orthic triangle is the orthocenter of the original triangle $ABC$.

Let $\Delta PQR$ be the pedal triangle of $\Delta ABC$. Excenters of $\Delta PQR$ are $(20,8), (4,12), (13,1)$.

If one of the vertices of $\Delta PQR$ is $(14,2),$ then find the area of $\Delta ABC$.

**Details and Assumptions:**

- Pedal triangle of a triangle is formed by joining feet of altitudes to the sides of the triangle.
- An excenter of a triangle is a point of intersection of an internal angle bisector and two external angle bisectors of the triangle.

Note: Try to solve this within a minute.

The orthic triangle also has the smallest perimeter among all triangles inscribed in an acute triangle $ABC$.

Finally, the orthic triangle is highly related to the tangential triangle, whose sides are the tangents to the circumcircle at the three vertices. The orthic and tangential triangle are homothetic