Question:
Let $A(1,0), B(6,2)$ and $C\left(\frac{3}{2}, 6\right)$ be the vertices of a triangle $\mathrm{ABC}$. If $\mathrm{P}$ is a point inside the triangle $\mathrm{ABC}$ such that the triangles APC, APB and BPC have equal areas, then the length of the line segment $\mathrm{PQ}$, where $\mathrm{Q}$ is the point $\left(-\frac{7}{6},-\frac{1}{3}\right)$, is__________.
Solution:
$P$ is centroid of the triangle $A B C$
$\Rightarrow P \equiv\left(\frac{17}{6}, \frac{8}{3}\right)$
$\Rightarrow P Q=5$
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