Question:
Let $A(1,0), B(6,2)$ and $\mathrm{C}\left(\frac{3}{2}, 6\right)$ be the vertices of a triangle $A B C$. If $P$ is a point inside the triangle $A B C$ such that the triangles $A P C, A P B$ and $B P C$ have equal areas, then the length of the line segment
$P Q$, where $Q$ is the point $\left(-\frac{7}{6},-\frac{1}{3}\right)$, is_________.
Solution:
P will be centroid of $\triangle A B C$
$P\left(\frac{17}{6}, \frac{8}{3}\right) \Rightarrow P Q=\sqrt{(4)^{2}+(3)^{2}}=5$