In an isosceles triangle, the apex angle is alpha, and the bisector of the base angle is beta.

In an isosceles triangle, the apex angle is alpha, and the bisector of the base angle is beta. Find the lengths of the sides of the triangle.

We denote the angle at the vertex ABC, the point of intersection of the bisector and with the side BC through M, then:

BAC = (180 – ABC) / 2 = (180 – α) / 2 = 90 – α / 2.

BAM = 1/2 * BAC = 45 – α / 2.

BMA = 180 – α – (45 – α / 2) = 135 + α / 2.

Consider a triangle ABM, by the sine theorem we get:

| AM | / sin (ABM) = | AB | / sin (BMA);

| AB | = | AM | * sin (BMA) / sin (ABM) = ß * sin (135 + α / 2) / sin (α).

Since the triangle is isosceles: | AB | = | BC |.

By the sine theorem for triangle ABC:

| AC | / sin (ABC) = | AB | / sin (BMA).