The Excenters and the Orthic Triangle
- Author:
- Julie Buasen
The excenter is the intersection of the bisectors of two exterior angles and that of the remaining interior angle. The excenter is the center of the excircle, the circle that is tangent to extensions of two sides and the remaining side.
The excenter is the intersection of the bisectors of two exterior angles and that of the remaining interior angle. The excenter is the center of the excircle, the circle that is tangent to extensions of two sides and the remaining side.
Consider the following:
- The altitudes and sides of triangle GHD are interior and exterior angle bisectors of orthic triangle ABC, so K is the incenter of triangle ABC and G, H, D are the 3 excenters
- The sides of the orthic triangle form an "optical" or "billiard" path reflecting off the sides of ABC.
- The orthic triangle ABC has the smallest perimeter of any triangle with vertices on the sides of GHD.
- The altitudes and sides of triangle GHD are interior and exterior angle bisectors of orthic triangle ABC, so K is the incenter of triangle ABC and G, H, D are the 3 excenters.