Question

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.

Answer 1

Let ABC be a right triangle right angled at P. 

Consider a circle with diametere 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 = QP …….. (1) 

Also, ∠QPC = ∠QCP 

∴ PQ = QC .......... (2) 

From (1) and (2); 

QB = QC 

Hence proved.