Let $ABC$ be an acute angled triangle whose incenter and centroid are respectively $I$ and $G$.$AI,BI$ and $CI$ cuts the sides of the triangle at $P,Q,R$ respectively.If $p_1,p_2$ and $p_3$ are the lengths of the altitudes through $A,B$ and $C$ respectively and $G$ lies on $PQ$ then prove by vector method that $p_1+p_2=p_3$.
I could not much do about this complex problem which i could mention here.I know that incenter,orthocenter and centroid all lie on one line