91Ó°ÊÓ

In a triangle \(A B C\), prove that, \(b^{2} \sin 2 C+c^{2} \sin 2 B=2 b c \sin A\)

Prove that the distance between the circum-center and the orthocenter of a triangle is \(O H=\) \(R \sqrt{1-8 \cos A \cdot \cos B \cdot \cos C}\)

Prove that the distance between the in-center and the ex-centers are $$ \begin{gathered} I I_{1}=4 R \sin \left(\frac{A}{2}\right), I I_{2}=4 R \sin \left(\frac{B}{2}\right) \\ I I_{3}=4 R \sin \left(\frac{C}{2}\right) \end{gathered} $$

The ex-radii \(r_{1}, r_{2}, r_{3}\) of a triangle \(A B C\) are in H.P., prove that the sides \(a, b, c\) are in A.P.

If the sides of a triangle are in A.P. and if its greatest angle exceeds the least angle by \(\alpha\), show that the sides areinthe ratio \((1-x): 1:(1+x)\), where \(x=\sqrt{\frac{1-\cos \alpha}{7-\cos \alpha}}\).