Question from http://www.sgforums.com/forums/2297/topics/359741
A line PQ is drawn through the verex A of triangle ABC such that BP and CQ are perpendicular to PQ. If M is the midpoint of BC and MR is perpendicular to PQ, prove that MP = MQ.
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Answer:
We can draw an auxiliary line (in red) to help us see easier.
Draw it such that CH is perpendicular to MR and CG perpendicular to BP
(Note: BP is parallel to MR and parallel to CQ since all are perpendicular to PQ)
We can see that triangle BCG and MCH are similar, with BM = MC
Hence, by intercept theorem, GH = HC
Since PGHR and RHCQ are rectangles, PR = GH = HC = RQ
PR = RQ
Since PR = RQ and MR is perpendicular to PQ, then triangle MPQ is isosceles.
Hence, MP = MQ (shown)
