$\vec{PP_1}+\vec{PP_2}+\vec{PP_3}=\frac {3}{2}\vec {PO} $. Let $ABC $ an equilateral  triangle and $P\in int (ABC) $. 
If $O $ is the center of gravity of  the triangle and $P_1, P_2, P_3$ are the projections of $P $ on the sides then 
$\vec{PP_1}+\vec{PP_2}+\vec{PP_3}=\frac {3}{2}\vec {PO} $.

Given the vertices $A,B,C$ then $O=\frac{A+B+C}3$ which is a weighted sum with equal weights. Now $P=xA+yB+zC$ for some weights $x,y,z$ such that $x+y+z=1.$ Suppose $P_1$ is the projection of $P$ on side $AB$ with midpoint $M=\frac{A+B}2$. Then $P_1-P= M-(xM+yM+zC)$ and similarly for the other two projections. Calculation shows that $\frac{P_1+P_2+P_3}3-P=\frac{O-P}2.$
Note that this proof uses affine geometry which works for any triangle but we need to interpret projection onto a side as parallel projection along the same direction as the median onto that side.
