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
Using trilinear coordinates, we have $I=[1;1;1]$ and $G=\left[\frac{1}{a},\frac{1}{b},\frac{1}{c}\right]=[h_a;h_b;h_c]$, so: $$ P=[0;1,1],\quad Q=[1;0;1] $$ and $G$ lies on $PQ$ iff $\det[G,P,Q]=0$. Since: $$\det\begin{pmatrix}h_a & h_b & h_c \\ 0 & 1 & 1 \\ 1 & 0 & 1\end{pmatrix}=h_a+h_b-h_c$$ the claim trivially follows.