# Geometry Problem 1575: Proving an Angle Bisector in a Triangle Involving an Altitude, Midpoint, and Excircle. A High School and College Challenge

In triangle ABC, let E be the midpoint of altitude AD from A to BC. The excircle opposite C is tangent to BC at G, and line EG intersects the excircle at H. Prove that HG bisects angle BHC.

In triangle's grace,
Midpoint, altitude, and excircle,
Bisectors reveal.

## Key Definitions and Descriptions

Key Term Description
Triangle The geometric figure ABC, consisting of three sides and three angles.
Altitude The perpendicular segment AD from vertex A to side BC in triangle ABC.
Midpoint The point E that divides the altitude AD into two equal segments.
Excircle The circle opposite vertex C that is tangent to side BC of triangle ABC at point G.
Tangency Point Point G where the excircle opposite C is tangent to BC.
Perpendicular The line segment AD, which is perpendicular to side BC of triangle ABC.
Intersection The point H where line EG intersects the excircle opposite C.
Angle Bisector The line HG that bisects angle BHC in the geometric problem.
Bisects Angle To divide an angle into two equal parts. In this problem, line HG bisects angle BHC.