In the triangle ABC with an obtuse angle BAC, the heights BB1 and CC1 are drawn.

In the triangle ABC with an obtuse angle BAC, the heights BB1 and CC1 are drawn. Prove that triangles AB1C1 and ABC are similar.

First, let us prove the similarity of triangles ABB1 and ACC1.

Both triangles are rectangular, the angle BAB1 = CAC1 as vertical angles, then the right-angled triangles ABB1 and ACC1 are similar in acute angle.

Then the segments AB1, AC1 and AB and AC are proportional.

In triangles AB1C1 and ABC, the angle B1AC1 = ABC as vertical angles, then the triangles AB1C1 and ABC are similar in two proportional sides and the angle between them, which was required to be proved.