Question

In the given figure, AP ⊥ BC, BR ⊥ AC and CQ ⊥ AB. Prove that ∠OPQ = ∠OPR.

Answer 1

As the two right triangles ABR and APB are on the same side of AB and which is also the hypotenuse of both , the circle drawn taking AB as diameter will pass through R and P. So ABPR is a cyclic quadrilateral.

Hence ∠ RPC =∠BAR

Similarly ACPQ is also cyclic quadrilateral

So   ∠ BPQ =∠QAC

But ∠BAR =∠QAC

So ∠ RPC=∠ BPQ

=>90- ∠ RPC=90-∠ BPQ

=> ∠ OPQ =∠OPR