(Q12) In a right triangle ABC, a circle with a side AB as diameter is drawn to intersect the hypotenuse AC in P. Prove that the tangent to the circle at P bisects the side BC.

ABCPQO

Let ABC be a right triangle right angled at P.

Consider a circle with diameter AB

From the figure, the tangent to the circle at B meets BC in Q

Now QB and QP are two tangents to the circle from the same point P

QB = -----(1)

Also ∠QPC = ∠

PQ = -------(2)

From (1) and (2)

QB = QC Hence proved