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Question

ABD is inscribed in a circle.Poit P lies on the circumscribed circle of a traingle be such that through point P. PN,PM and PL are perpendicular on sides of triangle (possibly by increasing sides).Prove that points N,M,L are collinear.

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Solution


Construction :- Join NM , ML and AP

PNA=90° (given)PMA=90° (given)Adding both we get,PNA+PMA=180° Since, the opposite angles of a quadrilateral are supplementary.Hence, PNAM is a cyclic quadrilateral.PAN=PMN....(1) (angles in the same segment)PABC is acyclic quadrilateral.PAN=PCB (Exterior angle of a cyclic quadrilateral is equal to its interior opposite angle)or, PAN=PCL...............(2)from equation (1) and (2)PMN=PCL........(3)PMC=PLC=90°Since, PC subtend equal angles at points M and L.Hence, PMLC is a cyclic quadrilateral.So, PCL+PML=180°PMN+PML=180° (using eqn 3)Also, PMN and PML are adjacent angles.Hence, points N, M and L are collinear.

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