Question

In ∆PQR seg PM is a median. Angle bisectors of ∠PMQ and ∠PMR intersect side PQ and side PR in points X and Y respectively. Prove that XY || QR.

Answer 1

In △PMQ, ray MX is bisector of △PMQ.
$\therefore \frac{\boxed{\mathrm{PX}}}{\boxed{\mathrm{XQ}}}=\frac{\boxed{\mathrm{MQ}}}{\boxed{\mathrm{MP}}}$ .......... (I) theorem of angle bisector.
In △PMR, ray MY is bisector of△PMR.
$\therefore \frac{\boxed{\mathrm{PY}}}{\boxed{\mathrm{YR}}}=\frac{\boxed{\mathrm{MR}}}{\boxed{\mathrm{MP}}}$ .......... (II) theorem of angle bisector.
But$\frac{\mathrm{MP}}{\mathrm{MQ}}=\frac{\mathrm{MP}}{\mathrm{MR}}$ ......... M is the midpoint QR, hence MQ = MR.
$\therefore \frac{\mathrm{PX}}{\mathrm{XQ}}=\frac{\mathrm{PY}}{\mathrm{YR}}$
∴XY || QR .......... converse of basic proportionality theorem.