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`.

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

In the given figure, a circle is inscribed in a DeltaABC such that it touches the sides AB, BC, and CA at points D, E, and F respectively. If the lengths of the sides AB, BC, and CA are 12 cm, 8 cm, and 10 cm respectively, find the lengths of AD, BE, and CF.

AD is an altiude of an isosceles triangle ABC in which AB = AC. Show that (i) AD bisects BC (ii) AD bisects A.

AD is an altitude of an isosceles triangle ABC in which AB = AC. Show that, (i) AD bisects BC (ii) AD bisects /_ A.

In an isosceles triangle ABC, if AB = AC = 25 cm and altitude from A on BC is 24 cm, find BC.

In the figure, the sides AB, BC and CA of triangle ABC touch a circle with centre O and radius r at P, Q , and R respectively. Prove that (i) AB+CQ=AC+BQ (ii) Area (DeltaABC)=1/2 ("Perimeter of" DeltaABC)xx r

In each of the following questions, we ask you to prove a statement. List all the steps in each proof, ang give the reason for each step. 6. A line parallel to side BC of a triangle ABC ,intersects AB and AC at D and E respectively. Prove that (AD)/(DB)=(AE)/(EC) .

ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively. Show that these altitudes are equal.

ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively. Show that these altitudes are equal.

In an isosceles triangle ABC with AB = AC, D and E are points on BC such that BE = CD (see figure) Show that AD = AE