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 \(ABC\).
The red triangle has a smaller perimeter than the green one.
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