How Can You Prove That Triangle BMP and CMQ Are Congruent in this Diagram?

[Styx]

In the diagram, AM is a medium of triangle ABC. Perpendicular lines drawn from B and C to AM (or its extension) meet AM at P and Q respectively.

Prove that BP = CQ

So far I have concluded that:

BM = CM
Angle BPM = angle CQM
Triangle ABM = ACM

I am not sure what else I can do in order to prove that the triangles BMP and CMQ are congruent which would prove that BP = CQ

 

[HallsofIvy]

Since the lines are drawn perpendicular to AM, you have RIGHT triangles. Further, since M is a "median" (note spelling) BM and CM are congruent.

 

[Styx]

Consider Triangles BMP and CMQ
You know

BM = CM (given hypothesis)

angle BPM = angle CQM as they are both at right angles to AM or its extension

180 degrees - angle AMC = angle AMB, angle AMB + angle QMB = 180 degrees
Therefore, angle AMC = angle AMB

Triangle BMP and CMQ are congruent by the AAS condition of the triangle theorem.

Therefore, BP = CQ