Set the bisector line DA to horizontal position and extend the triangle ABC in the opposite direction to get triangle AFE.

Because AD || BE (parallel lines). The triangles ADC ⁓ BCE are similar

Because AE = AB we get the general form of the angle bisector theorem.

Draw the bisector line from vertex C to the side AB and mark the intersection point as D.

Also mark the length of segment BD as y.

Notice that triangle ACD is isosceles whose side AD is equal to: AD = AB − y = x + 4 − y

According to the angle bisector theorem we have the relation

Realise that the triangles ACB and triangle CBD are similar triangles based on three equal angles.

Solving equations (1) and (2) for x and y we get the quadratic equation: x^{2} − 6x − 16 = 0

The solution of this equation is x = 8 (other solution x =-2 is irrelevant) and the sides of the triangle are 8, 10 and 12.