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.

Diagram of right-angled triangle ABC with altitude, bisectors, and midpoint proving collinearity. Geometry Problem 1563.

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

Academic Levels: Suitable for High School and College Mathematics Education

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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).