In a convex quadrilateral $ABCD$ rays $AB$ and $DC$ intersect at point $P$, and rays $BC$ and $AD$ at point $Q$. There is a point $T$ on the diagonal $AC$ such that the triangles $BTP$ and $DTQ$ are similar, in that order. Prove that $BD \Vert PQ$.
Source: 2016 239 J2
Tags: geometry
In a convex quadrilateral $ABCD$ rays $AB$ and $DC$ intersect at point $P$, and rays $BC$ and $AD$ at point $Q$. There is a point $T$ on the diagonal $AC$ such that the triangles $BTP$ and $DTQ$ are similar, in that order. Prove that $BD \Vert PQ$.