# Geometry Problem 1563: Proof of Perpendicularity in a Right Triangle involving the Altitude, Angle bisectors, and Midpoints.

In a right triangle ABC, with angle ABC being 90 degrees, let BH be the altitude from B to AC and BD the bisector of angle ABC. The bisector of angle AHB intersects AB at E, and the bisector of angle BHC intersects BC at F. Prove that BD is perpendicular to the line passing through the midpoints of HE and HF.

BD stands so tall,
GJ whispers in the lines,
Perpendicular!

## Key Definitions and Descriptions

Definition Description
Right Triangle ABC A right triangle where angle ABC is 90 degrees.
Altitude BH The line segment from point B perpendicular to line AC.
Angle Bisector BD The line segment from B that divides angle ABC into two equal angles.
Point E The intersection of the bisector of angle AHB with line AB.
Point F The intersection of the bisector of angle BHC with line BC.
Line HE The line segment connecting points H and E.
Line HF The line segment connecting points H and F.
Midpoints of HE and HF G and J, the points that are halfway along line segments HE and HF.
BD is Perpendicular to GJ Line segment BD intersects the line segment GJ at a right angle (90 degrees).