O is the orthocentre of △ABC if and only if AP⊥BC, BR⊥AC and CQ⊥AB. Prove that angle OPQ= angle OPR

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O is the orthocentre of △ABC if and only if AP⊥BC, BR⊥AC and CQ⊥AB. Prove that angle OPQ= angle OPR

Answer 1

This is only angle chasing, $\angle OPC=90^{\circ}$. Also, $\angle RPC =BAC$ because $ARPB$ is a cyclic quadrilateral. In the same way $\angle QPB = \angle BAC$. From this point, the conclusion follows.