Question

In ∆ABC, the altitudes from vertex B and C intersect at point M, so that BM = CM. Prove that ∆ABC is isosceles.

Answer 1

BM=CM implies that BCM is isosceles. This means that an altitude from the third vertex A of the triangle ABC, which must necessarily go through M will intersect BC exactly in the middle. This implies that the angles <CBA = <BCA which means the ABC is isosceles.