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:
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.