is right-angled if and only if![\[\sin A + \sin B + \sin C = \cos A + \cos B + \cos C + 1\]](https://latex.artofproblemsolving.com/e/c/c/eccbc0200d33117b14491e3a3e38f418beb09002.png)
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We know that $latex \sin A+\sin B+\sin C=\frac{s}{R}$ and $latex \cos A+\cos B+\cos C=1+\frac{r}{R}$,
hence $latex ABC$ is right-angled iff $latex \sin A+\sin B+\sin C=1+\cos A+\cos B+\cos C$.
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Prove that a triangle ABC is right-angled if and only if sin A+sin B+sin C= cos A+cos B+cos C+1;
<C=90;sin A+sin B+sin C=(a+b)/c+1; cos A+cos B+cos C=(a+b)/c+0=(a+b)/c;;
hence sin A+sin B+sin C= cos A+cos B+cos C+1 proved ;
how to show that if sin A+sin B+sin C= cos A+cos B+cos C+1 lead it to ABC as rt angled triangle?