In `DeltaPQR`, seg `XY||` side `QR`. `M` and `N` are the midpoints of seg `PY` and side `PR` respectively, `P-M-Y-N-R`.
Prove that `(i) DeltaPXM~DeltaPQN`
`(ii)` seg `XM||` seg `QN`.

Explanation

In `DeltaPXY` and `DeltaPQR`,
`/_PXY~=/_PQR`……(Seg `XY||` side `QR`, corresponding angles)
`/_XPY~=/_QPR`…..(Common angle)
`:.DeltaPXY~DeltaPQR`……("AA"test for similarity)
`:.(PX)/(PQ)=(PY)/(PR)`………..(c.s.s.t)…….`(1)`
Point `M` and `N` are the midpoints of seg `PY` seg `PR` respectively. .......(Given)
`:.PY=2PM` and `PR=2PN`........`(2)`
`:.(PX)/(PQ)=(2PM)/(2PN)`.....[From `(1)` and `(2)`]
`:.(PX)/(PQ)=(PM)/(PN)` ........`(3)`
In `DeltaPXM` and `DeltaPQN`,
`(PX)/(PQ)=(PM)/(PN)` [From `(3)`]
`/_XPM~=/_QPN`.......(Common angle)
`:.DeltaPXM~DeltaPQN` ......(SAS test for similarity)
`:./_PXM=/_PQN`........(c.a.s.t)
`:.` seg `XM||` seg `QN`.........(Corresponding angles test for parallel lines)