Draw median CD of \(\Delta\)ABC. Call E is the midpoint of  CG.

From D and E draw DH and EK perpendicular with line d.

We have GQ is the midsegment of trapezium HDEK.

\(\Rightarrow GQ=\dfrac{DH+EK}{2}\Leftrightarrow2GQ=DH+EK\)

\(\Rightarrow4GQ=2DH+2EK\) (1)

Consider trapezium MABN:

DH is midsegment \(\Rightarrow2DH=AM+BN\) (2)

Similar:  \(2EK=GQ+PC\) (3)

Substitute (2) and (3) to (1): \(AM+BN+GQ+PC=4GQ\)

\(\Rightarrow AM+BN+PC=3GQ\) .