# 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,

\[\angle AA'B' = AA'C', \angle BB'C' = \angle BB'A', \angle CC'A' = \angle CC'B'\]

which can be summarized as

The incenter of the orthic triangle is the orthocenter of the original triangle \(ABC\).

Let \(\Delta PQR\) be the pedal triangle of \(\Delta ABC\). Excentres 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 \(\Delta ABC\).

**Bonus**: Try to solve this within a minute.

**Details and Assumptions**

Pedal Triangle of a triangle is formed by joining foot of altitudes to a side of the triangle.

Excentres of a triangle are point of intersection of an internal angle bisector and two external angle bisectors of the triangle.

##### This problem is created by me.

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