Question

In a triangle ABC, $I$ be the incentre and $I_1,I_2$ and $I_3$ be the excentres. If R and r be the radius of circum-circle and in-circle respectively. Then $I_1{I}_2=$

• A. $2R\sin \frac{C}{2}$
• B. $4R\sin \frac{C}{2}$
• C. $2R\cos \frac{C}{2}$
• D. $4R\cos \frac{C}{2}$

Answer 1

The correct option is D $4R\cos \frac{C}{2}$

$I_1{I}_2=c\text{cosec}\frac{C}{2}{I}_1{I}_2=\frac{c}{\sin \frac{C}{2}}{I}_1{I}_2=\frac{2c\cos \frac{C}{2}}{2\sin \frac{C}{2}\cos \frac{C}{2}}{I}_1{I}_2=\frac{2c\cos \frac{C}{2}}{\sin C}\frac{c}{\sin C}=2R\Rightarrow I_1{I}_2=4R\cos \frac{C}{2}$