91Ó°ÊÓ

In a tringle \(\triangle A B C\), prove that \(\frac{\cos ^{2}\left(\frac{A}{2}\right)}{a}+\frac{\cos ^{2}\left(\frac{B}{2}\right)}{b}+\frac{\cos ^{2}\left(\frac{C}{2}\right)}{c}=\frac{s^{2}}{a b c}\)

In \(\Delta A B C\), prove that, \(a^{2}(s-a)+b^{2}(s-b)+c^{2}(s-c)\) \(=4 R \Delta\left(1-4 \sin \left(\frac{A}{2}\right) \sin \left(\frac{B}{2}\right) \sin \left(\frac{C}{2}\right)\right)\)

The ex-radii of a \(\Delta r_{1}, r_{2}, r_{3}\) are in A.P., then the sides \(a, b, c\) are in (a) A.P. (b) G.P. (c) H.P. (d) A.G.P.

In a \(\Delta A B C\), prove that \(\left(b c \cos ^{2}\left(\frac{A}{2}\right)+c a \cos ^{2}\left(\frac{B}{2}\right)+a b \cos ^{2}\left(\frac{C}{2}\right)\right)\) \(=\frac{1}{4}(a+b+c)^{2}\)

Let \(O\) be a point inside a triangle \(A B C\) such that \(\angle O A B=\angle O B C=\angle O C A=\omega\), then prove that (i) \(\cot A+\cot B+\cot C=\cot \omega\) (ii) \(\operatorname{cosec}^{2} A+\operatorname{cosec}^{2} B+\operatorname{cosec}^{2} C=\operatorname{cosec}^{2} \omega\)

In any \(\Delta A B C, \sum \frac{\sin ^{2} A+\sin A+1}{\sin A}\) is always greater than (a) 9 (b) 3 (c) 27 (d) 36

In a \(\Delta A B C\), prove that, \(\begin{aligned}(b-c) \cot \left(\frac{A}{2}\right)+(c-a) \cot \left(\frac{B}{2}\right) & \\ &+(a-b) \cot \left(\frac{C}{2}\right)=0 \end{aligned}\)

Find the distance between the circum-center and the mid-points of the sides of a triangle.

If the sides \(a, b, c\) of a triangle are in A.P., then find the value of \(\tan \left(\frac{A}{2}\right)+\tan \left(\frac{C}{2}\right)\) in terms of \(\cot \left(\frac{B}{2}\right)\)

In a \(\Delta A B C, a=2 b\) and \(|a-b|=\frac{\pi}{3}\), then \(\angle C\) is (a) \(\frac{\pi}{4}\) (b) \(\frac{\pi}{3}\) (c) \(\frac{\pi}{6}\) (d) None