RIOPLATENSE 2013 Problem 2

Let $ABCD$ be a square, and let $E$ and $F$ be points in $AB$ and $BC$ respectively such that $BE=BF$ . In the triangle $EBC$ , let N be the foot of the altitude relative to $EC$ . Let $G$ be the intersection between $AD$ and the extension of the previously mentioned altitude. $FG$ and $EC$ intersect at point $P$ , and the lines $NF$ and $DC$ intersect at point $T$ . Prove that the line $DP$ is perpendicular to the line $BT$ .