Answer: By the angle bisector theorem \(\frac{PX}{XQ} = \frac{MP}{MQ}\) and \(\frac{PY}{YR} = \frac{MP}{MR}\); since MQ = MR these are equal, so XY ∥ QR.
In \(\triangle PMQ\), ray MX bisects \(\angle PMQ\):
\[\frac{MP}{MQ} = \frac{PX}{XQ}\quad \text{(I)}\]In \(\triangle PMR\), ray MY bisects \(\angle PMR\):
\[\frac{MP}{MR} = \frac{PY}{YR}\quad \text{(II)}\]M is the midpoint of QR, so MQ = MR, hence \(\frac{MP}{MQ} = \frac{MP}{MR}\).
From (I) and (II), \(\frac{PX}{XQ} = \frac{PY}{YR}\).
\(\therefore\) In \(\triangle PQR\), XY ∥ QR (converse of basic proportionality theorem).
Boxes: \(\frac{MP}{MQ} = \frac{PX}{XQ}\); \(\frac{MP}{MR} = \frac{PY}{YR}\).