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Challenging geometry problem: similar triangles

  1. M is the midpoint of side BC of ∆ABC. P is any point on median AM. Extend BP and CP to meet sides AC and AB at D and E respectively.
    Prove that DE is parallel to BC.


     
  2. In isosceles triangle ABC with AB = AC, AD is the altitude to side BC. P is any point on AD, . Draw CF parallel to AB. Extend BP to intersect AC at E, CF at F.
    Prove that BP2 = PE·PF


     
  3. In ∆ABC, AD bisects angle A, meeting side BC at D.
    Prove that AD2 = AB·AC - BD·DC