The height BD is drawn in triangle ABC (point D lies on side AC). It turned out that, 2AB = CD

The height BD is drawn in triangle ABC (point D lies on side AC). It turned out that, 2AB = CD and 2CB = AD. Find the angles of triangle ABC.

Let CD = x, then AB will be equal to 2x. Let AD = y, then BC will be equal to 2y.

Triangle ABD is rectangular, we express the side BD according to the Pythagorean theorem:

BD² = AB² – AD² = (2x) ² – y² = 4x² – y².

The triangle BCD is rectangular, we express the side BD by the Pythagorean theorem:

BD² = ВС² – CD² = (2y) ² – x² = 4y² – x².

So 4x² – y² = 4y² – x².

4x² + x² = 4y² + y².

5x² = 5y².

x² = y².

Consequently, x = y, that is, AD = CD = x = y. Hence, AC = 2x = 2y.

Hence, AB = BC = AC.

Therefore, triangle ABC is equilateral and all angles of the triangle are equal to 60 °.