The incircle of an isosceles triangle `ABC`, with `AB=AC`, touches the sides `AB`, `BC`, `CA` at `D,E` and `F` respectively. Prove that `E` bisects `BC`.

The incircle of an isoceles triangle ABC, with AB=AC, touches the sides AB,BC and CA at D,E and F respecrively. Prove that E bisects BC.

The incircle of an isoceles triangle ABC, with AB=AC, touches the sides AB,BC and CA at D,E and F respecrively. Prove that E bisects BC.

The incircle of an isosceles triangle ABC in which AB=AC , touches the sides BC, CA, and AB at D, E, and F respectively. Prove that BD=DC .

The incircle of an isosceles triangle ABC, in which AB = AC, touches the sides BC, CA and AB at D, E and F respectively . Prove that BD = DC.

The incircle of a triangle ABC touches the sides AB, BC and CA at the points F, D and E resepectively. Prove that AF + BD + CE + DC + EA = 1/2 (Perimeter of triangle ABC)

An isosceles triangle ABC, with AB = AC, circumscribes a circle, touching BC at P, AC at Q and AB at R. Prove that the contact point P bisects BC.

Let's denote the semi-perimeter of a triangle ABC in which BC = a, CA = 5, AB = c. If a circle touches the side BC, CA, AB at D, E, F respectively, prove that BD = S-b.

The sides AB and AC are equal of an isosceles triangle ABC. D E and F are the mid-points of sides BC, CA and AB respectively. Prove that: (i) Line segment AD is perpedicular to line segment EF. (ii) Line segment AD bisects the line segment EF.