Three forces $\vec{P}, \vec{Q}$ and $\vec{R}$ acting along IA, IB and IC, where I is the incentre of a…

Three forces $\vec{P}, \vec{Q}$ and $\vec{R}$ acting along IA, IB and IC, where I is the incentre of a $\triangle A B C$, are in equilibrium. Then $\vec{P}: \vec{Q}: \vec{R}$ is
  1. $\cos \frac{A}{2}: \cos \frac{B}{2}: \cos \frac{C}{2}$
  2. $\sin \frac{A}{2}: \sin \frac{B}{2}: \sin \frac{C}{2}$
  3. $\sec \frac{A}{2}: \sec \frac{B}{2}: \sec \frac{C}{2}$
  4. $\operatorname{cosec} \frac{A}{2}: \operatorname{cosec} \frac{B}{2}: \operatorname{cosec} \frac{C}{2}$

Solution

By Lami's theorem $ \begin{aligned} & \vec{P}: \vec{Q}: \vec{R}=\sin \left(90^{\circ}+\frac{A}{2}\right): \sin \left(90^{\circ}+\frac{B}{2}\right): \sin \left(90^{\circ}+\frac{C}{2}\right) \\ & \Rightarrow \cos \frac{A}{2}: \cos \frac{B}{2}: \cos \frac{C}{2} . \end{aligned} $

Asked in: JEE Main 2004

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