Geometry problem on the incentre and circumcenter of a triangle I have the following problem:


  In a triangle $ABC$ the line joining incentre and circumcentre is parallel to side $BC$. Prove that $\cos B + \cos C=1$.


Could someone help me solve it?
This question provides simple proofs of the identity: $$\frac{r}{R}+1=\cos A+\cos B+\cos C$$

Referring to Derek's diagram, $\angle CKB = 2\angle A = 2\angle CKR$. Since, the angle made by the chord $BC$ at the point $A$ on the circumcircle of $\Delta ABC$ is half the angle made at the circumcentre $K$.

Also, $\cos A = \cos \angle CKR = \dfrac{KR}{CK}=\dfrac{r}{R}$ (since, IK is parallel to $BC$, we have $KR = r$).

So, $\cos B + \cos C = 1$
Given Line joining Circumcentre and Incentre are parallel
$$r=4R\sin A/2\sin B/2\sin C/2=R\cos A$$
$$4\sin A/2\sin B/2\sin C/2=\cos A$$
$$\cos A+\cos B+\cos C-1=\cos A$$
$$\cos B+\cos C=1$$