# 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.

****PROVE THAT THE ALTITUDES ARE ANGLE BISECTORS OF THE ORTHIC TRIANGLE.**