Question

PQRS is quadrilateral whose vertices lie on the circumference of a circle and if M is a point on line segment PR such that PM=QM and QMS is a straight line prove that MS=MR

Answer 1

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In triangles PQS and PQR

∠P is common to both.

and ∠PSQ=∠PQR

Triangles PQS and PQR are equiangular

∴PQPS=QRQS

or, PS=QRPQ⋅QS ...(i)

Again, triangles QRS and PQR are equiangular

∴QRSR=PQQS

or, SR=PQQS⋅QR ...(ii)

From eqns. (i) and (ii)

SRPS=QRPQ⋅QS⋅QR⋅QSPQ=QR2PQ2