In triangle ABC, the median BB1 is the bisector of the angle ABC. Prove that AB = CB.

Let’s make an additional construction. On the continuation of the segment BB1, we mark the segment B1M = BB1.

In triangles ABB1 and CMB1 AB1 = CB1 since BB1 ​​is the median, BB1 = MB1 by construction, the angle AB1B = CB1M is vertical, then triangles ABB1 and CMB1 are equal on two sides and the angle between them.

Then the angle ABB1 = CMB1 as similar angles of equal triangles.

Since the angle ABB1 = CBB1 since BB1 ​​is a bisector, and the angle ABB1 = CMB1, then the angle CBB1 = CMB1.

Then in the BCM triangle the angles at the base of the BM are equal, then the BCM triangle is isosceles, CM = BC, and since CM = AB, then AB = BC, and the ABC triangle is isosceles, which was required to prove.