If the triangle ABC is oblique (not right-angled), the points of intersection of the altitudes with the sides of the triangle form another triangle, A'B'C', called the orthic triangle or altitude triangle. It is the pedal triangle of the orthocenter of the original triangle. Also, the incenter (that is, the center for the inscribed circle) of the orthic triangle is the orthocenter of the original triangle.
The orthic triangle is closely related to the tangential triangle, constructed as follows: let LA be the line tangent to the circumcircle of triangle ABC at vertex A, and define LB and LC analogously. Let A" = LB ∩ LC, B" = LC ∩ LA, C" = LC ∩ LA. The tangential triangle, A"B"C", is homothetic to the orthic triangle.
The orthic triangle provides the solution to Fagnano's problem, posed in 1775, of finding for the minimum perimeter triangle inscribed in a given acute-angle triangle.
The orthic triangle of an acute triangle gives a triangular light route.
Trilinear coordinates for the vertices of the orthic triangle are given by
- A' = 0 : sec B : sec C
- B' = sec A : 0 : sec C
- C' = sec A : sec B : 0
Trilinear coordinates for the vertices of the tangential triangle are given by
- A" = −a : b : c
- B" = a : −b : c
- C" = a : b : −c
Read more about this topic: Altitude (triangle)