91Ó°ÊÓ

PROVING IDENTITIES RELATED TO EX-RADII $$ R=\frac{\left(r_{1}+r_{2}\right)\left(r_{2}+r_{3}\right)\left(r_{3}+r_{1}\right)}{4\left(r_{1} r_{2}+r_{2} r_{3}+r_{3} r_{1}\right)} $$

Prove that the area of any quadrilateral is one-half the product of the two diagonals and the sine of the angle between them.

The sides of a quadrilateral with an inscribed circle are \(7,10,5\) and \(2 \mathrm{~cm}\). and the sum of a pair of opposite angles is \(120^{\circ}\). Find area and radius of inscribed circle.

Given that the area of a polygon of \(n\) sides circumscribed about a circle is to the area of the circumscribed polygon of \(2 n\) sides as \(3: 2\), find \(n\).