# Thread: Concurrency at interior point

1. ## Concurrency at interior point

Let $\displaystyle P$ be an interior point of $\displaystyle \triangle ABC$such that the lines $\displaystyle AA_{1},BB_{1},CC_{1}$ are concurrent at $\displaystyle P$ and the points $\displaystyle A_{1},B_{1},C_{1}$ lie on $\displaystyle BC,CA,AB$ respectively.Find using vectors or otherwise value of
$\displaystyle \frac{PA_{1}}{AA_{1}}+\frac{PB_{1}}{BB_{1}}+\frac{ PC_{1}}{CC_{1}}$

2. Let $\displaystyle PQ\perp BC, \ AD\perp BC$

Then, $\displaystyle \frac{PA_1}{AA_1}=\frac{PQ}{AD}$

$\displaystyle A_{\Delta PBC}=\frac{PQ\cdot BC}{2}, \ A_{\Delta ABC}=\frac{AD\cdot BC}{2}$

$\displaystyle \frac{A_{\Delta PBC}}{A_{\Delta ABC}}=\frac{\frac{PQ\cdot BC}{2}}{\frac{AD\cdot BC}{2}}=\frac{PQ}{AD}$

So, $\displaystyle \frac{PA_1}{AA_1}=\frac{A_{\Delta PBC}}{A_{\Delta ABC}}$

In a similar way we have

$\displaystyle \frac{PB_1}{BB_1}=\frac{A_{\Delta PAC}}{A_{\Delta ABC}}$ and $\displaystyle \frac{PC_1}{CC_1}=\frac{A_{\Delta PAB}}{A_{\Delta ABC}}$

Then $\displaystyle \frac{PA_1}{AA_1}+\frac{PB_1}{BB_1}+\frac{PC_1}{CC _1}=\frac{A_{\Delta PBC}+A_{\Delta PAC}+A_{\Delta PAB}}{A_{\Delta ABC}}=\frac{A_{\Delta ABC}}{A_{\Delta ABC}}=1$