Prove that if the acute angles of a right-angled triangle are 1: 3, then the bisector of the largest angle

Prove that if the acute angles of a right-angled triangle are 1: 3, then the bisector of the largest angle is equal to one of the legs.

Let us denote by A = 90, B – the larger angle, BM – the bisector.

The tangent of angle B will be equal to the ratio of the legs, that is, 3: 1 = 3 (according to the condition of the problem), then the angle A = 60.

Since BM is a bisector, the angle ABM is:

60: 2 = 30.

Then:

AM = AD * tg (30) = AD / √3

BM = √ (AD) ^ 2 + AD ^ 2/3 = AD * 2 / √3

Since 3 * AD = AC, BM = AC.