Let S denote the semiperimeter of a triangle ABC in which BC = a, CA = b, AB = c. If a circle touches the sides BC, CA, AB at D, E, F, respectively. Prove that BD = s-b.

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.

Let s denotes the semi-perimeter of a DeltaABC in which BC=a, CA=b and AB=c. If a circle touches the sides BC, CA, AB, at D, E, F, respectively. Prove that BD=s-b.

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 , 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.

Let ABC and A'B'C' be two triangles in which AB gt A'B', BC gt B'C' and CA gt C'A' . Let D, E and F be the midpoints of the sides BC, CA and AB respectively. Let D', E' and F' be the midpoints of the sides B'C', C'A' and A'B' respectively. Consider the following statements : Statement I: AD gt A'D', BE gt B'E' and CF gt C'F' are always true. Statement II : (AB^(2)+BC^(2)+CA^(2))/(AD^(2)+BE^(2)+CF^(2)) =(A'B'^(2)+B'C'^(2)+C'A'^(2))/(A'D'^(2) +B'E'^(2) +C'F'^(2)) Which one of the following is correct in respect of the above statements?

Find the area of triangle ABC , the midpoints of whose sides AB, BC and CA are D(3,-1), E(5, 3) and F(1,-3) respectively.