2016 AMC 10A Problems/Problem 19

In rectangle ABCD, $AB=6$ and $BC=3$ . Point $E$ between $B$ and $C$ , and point $F$ between $E$ and $C$ are such that $BE=EF=FC$ . Segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BD}$ at $P$ and $Q$ , respectively. The ration $BP:PQ:QD$ can be written as $r:s:t$ where the greatest common factor of $r,s$ and $t$ is 1. What is $r+s+t$ ?

$\textbf{(A) } 7 \qquad \textbf{(B) } 9 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 15 \qquad \textbf{(E) } 20$

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2016 AMC 10A Problems/Problem 19