Question

triangle ABC is a right triangle right angled at A such that a B is equals to AC and bisector of angle C intersect the side AB at D prove that AC + AD is equals to Bc.

Answer 1

Let E be the foot of a perpendicular from D to BC.

Then triangle ACD is congruent to triangle ECD (since they have two corresponding angles equal and a common side). Therefore AD = DE and AC = CE.

Also, triangle BDE is an isosceles right triangle (angle DBE is 45° since triangle ABC is an isosceles right triangle). Therefore DE = EB.

Thus, AC + AD = CE + DE = CE + EB = BC.