Question

If in a triangle $ABC$, the line joining the circumcentre $O$ and the incentre $I$ is parallel to $BC$, then

• A. $r=R\cos A$
• B. $r=R\sin A$
• C. $R=r\cos A$
• D. $R=r\sin A$

Answer 1

The correct option is A

$r=R\cos A$

Explanation for the correct option.

Find the correct relation.

The given condition that the line joining the circumcentre $O$ and the incentre $I$ is parallel to $BC$ can be shown in the figure as:

As $OI$ is parallel to $BC$, so $O$ and $I$ are equidistant from $BC$ and thus $OL=IM=r\mathbf{}\left(OL\perp BC,IM\perp BC\right)$

Now, as $O$ is the circumcenter so $\angle BOC=2A$ and also

$\begin{array}{rcl}\angle BOL& =& \frac{1}{2}\angle BOC\\ & =& \frac{1}{2}\times 2A\\ & =& A\end{array}$

Again as $OB$ is the orthoradius, so $OB=R$.

Now, in $△OBL$,

$$\begin{gathered} \cos A=\frac{OL}{OB} \\ \Rightarrow \cos A=\frac{r}{R} \\ \Rightarrow R\cos A=r \end{gathered}$$

Hence, the correct option is A.