Similarity in chaos

Let the triangle be $ABC$ , where $BC=6$ . Then we note that $\triangle APQ$ and $\triangle ABC$ are similar. Let the altitude from $A$ to $BC$ be $h$ and the inradius be $r$ . Then the altitude from $A$ to $PQ$ is $h-r$ , and

$\begin{aligned} \frac {PQ}{BC} & = \frac {h-r}h \\ \implies PQ & = 6 \times \frac {h-r}h & \small \blue{\text{Since area of }\triangle ABC, A = \frac 12 \times 6h = \frac {6+7+8}2 r} \\ & = 6 \times \frac {\frac A3 - \frac {2A}{21}}{\frac A3} \\ & = 6 \times \frac 57 = \boxed{\frac {30}7} \end{aligned}$