In triangle ABC,if the line joining the circu | vedclass

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(C) Let the circumcentre be $O$ and the incentre be $I$. The coordinates of $O$ and $I$ relative to the side $BC$ can be analyzed. The distance of $O$

Vedclass · Accepted Answer

$1$

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(C) Let the circumcentre be $O$ and the incentre be $I$. The coordinates of $O$ and $I$ relative to the side $BC$ can be analyzed. The distance of $O$ from $BC$ is $R \cos A$ and the distance of $I$ from $BC$ is $r$. Since $OI$ is parallel to $BC$,their distances from $BC$ must be equal,so $R \cos A = r$. Using the identity $r = 4R \sin(A/2) \sin(B/2) \sin(C/2)$,we have $\cos A = 4 \sin(A/2) \sin(B/2) \sin(C/2)$. Using $\cos A = 1 - 2 \sin^2(A/2)$,we get $1 - 2 \sin^2(A/2) = 4 \sin(A/2) \sin(B/2) \sin(C/2)$. Also,using the property $\cos B + \cos C = 2 \cos((B+C)/2) \cos((B-C)/2) = 2 \sin(A/2) \cos((B-C)/2)$. Given $OI \parallel BC$,it is a known property that $\cos B + \cos C = 1$.

In $\triangle ABC$,if the line joining the circumcentre $(O)$ and the incentre $(I)$ is parallel to $BC$,then $\cos B + \cos C = $

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