For $\vartriangle ABC$, let $D$ and $E$ be points on line segments $AB$ and $AC$ respectively such that $DE$ is parallel to $AB$. Suppose that $BD$ and $CE$ meet at $F$, and the tangents to the circumcircles of $\vartriangle ADF$ and $\vartriangle AEF$ at $D$ and $E$ respectively meet at $X$. Prove that there exist points $P, Q, R, S, T$ on lines $AF$, $AX$, $AB$, $BC$, $CA$ respectively such that (1) $PR + QR = PS + QS = PT + QT$ and (2) $AS, BT$ and $CR$ are concurrent.