ΔABC is an isosceles triangle in which. A circle touching all the three sides of ΔABC touches at D. Prove that D is the mid – point of.

Given that ΔABC is an isosceles triangle in which AB ≅ AC and a circle touching all the three sides of ΔABC touches BC at D.

We have to prove that D is the midpoint of BC.

Proof:

We know that if a circle touches all sides of a triangle, the lengths of tangents drawn from each vertex are equal.

∴ AE = AF, BD = BF and CD = CE … (1)

Consider AB = AC,

Subtracting AF from both sides,

⇒ AB – AF = AC – AF

From (1),

⇒ AB – AF = AC – AE

Since A – F – B and A – E – C,

⇒ BF = CE

From (1),

⇒ BD = CD

Since B – D – C and BD = CD,

D is the midpoint of BC.