Exam

Olympiad workbook

Subject

IOQM

Chapter

Vector Algebra

Olympiad workbook · IOQM · Vector Algebra

Let D, E, F be points on the sides B C, C A, A B of a triangle A B...

Let $D, E, F$ be points on the sides $B C, C A, A B$ of a triangle $A B C$, respectively. Suppose $A D, B E, C F$ are concurrent at $P$. If $P F / P C=2 / 3, P E / P B=2 / 7$ and $P D / P A=m / n$, where $m, n$ are positive integers with $\operatorname{gcd}(m, n)=1$, find $m+n$.

Step-by-step solution

$$\begin{aligned} & \because \frac{P F}{P C}=\frac{2}{3} \\ & \Rightarrow \quad \frac{P F}{C F}=\frac{2}{5}=\frac{\Delta_1+\Delta_3}{\Delta}...(i) \end{aligned}$$
Similarly $\quad \frac{2}{9}=\frac{\Delta_2+\Delta_6}{\Delta}$...(ii)$and \quad \frac{m}{m+n}=\frac{\Delta_4+\Delta_5}{\Delta}$...(iii) Adding (i), (ii) and (iii); we get $$\begin{aligned} & \frac{2}{5}+\frac{2}{9}+\frac{m}{m+n}=1 \\ & \Rightarrow \frac{m}{m+n}=1-\frac{28}{45} \\ & \Rightarrow \frac{m}{m+n}=\frac{17}{45} \\ & \Rightarrow 28 m=17 n \\ & \Rightarrow \frac{m}{n}=\frac{17}{28} \\ & \Rightarrow m+n=45 \end{aligned}$$