Question

In triangle ABC; angle ABC = 90^o and P is a point on AC such that ∠PBC = ∠PCB.
Show that: PA = PB.

Answer 1

Let PBC = PCB = x
In the right angled triangle ABC,
∠ABC = 90°
∠ACB = x
⇒ ∠BAC = 180° - ( 90° + x )
⇒  ∠BAC = ( 90°- x )             ...(i)

and
∠ABP = ∠ABC - ∠PBC
⇒  ∠ABP = 90° - x                   ...(ii)

Therefore in the triangle ABP;
 ∠BAP = ∠ABP
Hence, PA = PB     ...[sides opp. to equal angles are equal]