Question

The distance between ex-centre $I_1$ opposite to A and circum-centre $O$ is $O{I}_1=$

• A. $R\sqrt{1+8\cos \frac{A}{2}.\sin \frac{B}{2}.\sin \frac{C}{2}}$
• B. $R\sqrt{1+8\cos \frac{A}{2}.\cos \frac{B}{2}.\cos \frac{C}{2}}$
• C. $R\sqrt{1+8\sin \frac{A}{2}.\sin \frac{B}{2}.\sin \frac{C}{2}}$
• D. $R\sqrt{1+8\sin \frac{A}{2}.\cos \frac{B}{2}.\cos \frac{C}{2}}$

Answer 1

The correct option is D $R\sqrt{1+8\sin \frac{A}{2}.\cos \frac{B}{2}.\cos \frac{C}{2}}$

Distance between orthocentre $\left(O\right)$ and incentre $\left(I\right)$ is

$O^{\prime }{I}^{\prime }=R\sqrt{1-8\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}}=\sqrt{R^2-2Rr}$

Distance between excentres and orthocentre is

$O^{\prime }{I}_1^{\prime }=R\sqrt{1+8\sin \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}}=\sqrt{R^2+2R{r}_1}$

$O^{\prime }{I}_2^{\prime }=R\sqrt{1+8cos\frac{A}{2}\sin \frac{B}{2}cos\frac{C}{2}}=\sqrt{R^2+2R{r}_2}$

$O^{\prime }{I}_3^{\prime }=R\sqrt{1+8cos\frac{A}{2}cos\frac{B}{2}\sin \frac{C}{2}}=\sqrt{R^2+2R{r}_3}$