Inscribe Right Triangle (Posted on 2010-07-18)

Let Q be a point on side BC of triangle ABC.

Construct points P and R on sides AB and AC
respectively such that PQR is a right angle
and PR is parallel to BC.

	Submitted by Bractals
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Construct point O as the midpoint of side BC. Construct circle K having BC as a diameter. Ray AQ intersects circle K at point D with Q between A and D. Join O wth A. Construct a line through Q parallel to OD intersecting OA at point E. Construct a line through E parallel to BC intersecting sides AB and AC in points P and R respectively. Join P with Q and Q with R. Proof: Similar triangles AER ~ AOC and AOD ~ AEQ: |AE| |AO| |AO| |AE| ------ = ------ = ------ = ------. |ER| |OC| |OD| |EQ| Therefore, |ER| = |EQ|. Similar triangles AEP ~ AOB and AOD ~ AEQ: |AE| |AO| |AO| |AE| ------ = ------ = ------ = ------. |EP| |OB| |OD| |EQ| Therefore, |EP| = |EQ| = |ER|. Therefore, Q lies on a circle with PR as a diameter. Thus, PQR is a right angle.

	Subject	Author	Date
	Thoughts	Brian Smith	2010-07-20 15:39:58