Question

Any point X inside $△DEF$ Is joinied to its vertices From a point P in DX, PQ is drawn parallel to DE meeting XE at Q and QR is drawn parallel to EF meeting XF At R Prove that $PR\parallel DF$

Answer 1

A $△DEF$ and a point X inside it X is joined t the vertices D,E and F P is a point on DX $PQ\parallel DE$ and $QR\parallel EF$ We shall prove $PR\parallel DF$ join PR
PROOF :- In $△XED$ $PQ\parallel DE$
$\frac{XP}{PD}=\frac{XQ}{QE}$ ......(i)(Using Theles Theorem)
again in $△XEF$ $QR\parallel EF$
$\frac{XQ}{QE}=\frac{XR}{RF}$ (Using Thales Theoram)
From (I) and (ii) we get
$\frac{XR}{PD}=\frac{XR}{RF}$
$\therefore In△XFD$ points R and P divide sides XF and XD in the same ratio so using The converse of Thales ( Basic Proportionality) Theorem we have $PR\parallel DF$