Question

If a line $L$ intersects three sides $BC,CA$ and $AB$ of $△ABC$ in points $P,Q$ and $R$ then $\frac{BP}{PC}\cdot \frac{CQ}{QA}\cdot \frac{AR}{RB}=$.

• A. $-1$
• B. $1$
• C. $2$
• D. $-2$

Answer 1

The correct option is A $-1$

Consider a $\Delta ABC$ and it is given that the line intersect three sides BC, CA and AB of $\Delta ABC in points P, Q and R then

Construct perpendiculars AD, RG and QI from A, R, Q on BC.

The $\Delta BRG$ and $\Delta ABH$ are similar because both are right angled and on same base.

Hence , $\frac{AB}{RB}=\frac{AH}{RG}$ ______ (i)

Similarly $\Delta QIC\sim \Delta AHC$

$\frac{CQ}{AC}=\frac{QI}{AH}$ ______ (ii)

$\Delta QIP\sim \Delta RPG$

$\frac{PR}{PQ}=\frac{RG}{QI}$ _______ (iii)

Multiplying (i), (ii) & (iii)

$\frac{AB}{RB}\times \frac{CQ}{AC}\times \frac{PR}{PQ}=1$