Any point $X$ inside a $△ABC$ is joined to the vertices. From a point $P$ on $AX$, $PQ$ is drawn parallel to $AB$ meeting $XB$ at $Q$ and $QR$ is drawn parallel to $BC$ meeting $XC$ at $R$. Prove that $PR\parallel AC$.

$ABCD$ is a quadrilateral; $P, Q, R$ and $S$ are the points on the sides $AB, BC, CD$ and $DA$ respectively such that $AP:PB=AS:SD=CQ:QB=CR:RD$. Prove that $PQRS$ is a parallelogram

On three-line segments $OA, OB$ and $OC$ points $L, M, N$ respectively are so chosen that $LM\parallel AB$ and $MN\parallel BC$ but neither $L, M, N$ nor $A, B, C$ are collinear. Show that $LN\parallel AC$.

$ABCD$ is a parallelogram and $P$ is a point on the side $BC$. $DP$ when produced meets $AB$ produced at $L$, Prove that $\frac{DP}{LP}=\frac{DC}{BL}$

$ABCD$ is a parallelogram and $P$ is a point on the side $BC$. $DP$ when produced meets $AB$ produced at $L$, Prove that $\frac{DL}{DP}=\frac{AL}{DC}$

In a triangle $△ABC$, $D$ and $E$ are points on the sides $AB$ and $AC$ respectively, such that $AD\times EC=AE\times DB$. Prove that $DE\parallel BC$  

The side $BC$ of a $△ABC$  is bisected at $D$, $O$ is any point on $AD$. $BO$ and $CO$ produced meet $AC$ and $AB$ at $E$ and $F$ respectively and $AD$ is produced to $X$ so that $D$ is the midpoint of $OX$. Prove that $AO:AX=AF:AB$ and show that $FE\parallel BC$.