Vertex on a Line (Posted on 2010-07-11)

Let ABC be a triangle with P a point on line BC
such that B lies between P and C.

Construct a line through P, intersecting sides
AB and AC in points Q and R respectively, such 
that vertex S of parallelogram QARS lies on 
line BC.

	Submitted by Bractals
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Solution:	(Hide)
Construct a semicircle with CP as a diameter (opposite the triangle so as not to confuse the figure). Construct a line through B perpendicular to CP intersecting the semicircle at point T. Construct point S on side BC such that |PS| = |PT|. Construct a line through S parallel to AC intersecting AB at point Q. Construct line PQ that intersects side AC in point R. Proof: We only need to prove that RS is parallel to AB. From similar triangles PBT and PTC, |PB| |PT| ------ = ------ |PT| |PC| Therefore by construction, |PB| |PS| ------ = ------ (1) |PS| |PC| From similar triangles PQS and PRC, |PS| |PQ| ------ = ------ (2) |PC| |PR| Combining equations (1) and (2), |PB| |PQ| ------ = ------ |PS| |PR| Therefore, triangles PBQ and PSR are similar. Thus, RS is parallel to QB and therefore to AB.

	Subject	Author	Date
	re: Solution	Harry	2010-07-13 20:00:54
	Solution	Brian Smith	2010-07-13 12:45:12
	re: Is this right? (spoiler)	Bractals	2010-07-12 18:20:43
	Is this right? (spoiler)	Corey	2010-07-12 16:51:56