Question 124
Let $ABC$ be a triangle and let $M$, $N$ be points on the sides $[AB]$ and $[BC]$, respectively, such that $\frac{AM}{MB}=\frac{m}{n}$ and $\frac{BN}{NC}=\frac{n}{p}$, where $m, n, p$ are positive real numbers with the property $p^{2}=mn$. Denote by $P$ the intersection of the lines $CM$ and $AN$. Show that $n \overrightarrow{PA}+m \overrightarrow{PB}+p \overrightarrow{PC}=\overrightarrow{0}$.