3. (a) In any triangle ABC , prove that $$\operatorname { Cot } \mathrm { A } / \mathrm { b } + \cot \mathrm { B } / 2 + \cot \mathrm { C } / 2 = \mathrm { A } / 2 \cot \mathrm {~B} / 2 \cot \mathrm { C } / 2$$ (b) Let ABC be a triangle with incentre I and inradius r . Let $\mathrm { D } , \mathrm { E } , \mathrm { F }$ be the feet of the perpendiculars from I to the sides $\mathrm { BC } , \mathrm { CA }$ and AB respectively. If $\mathrm { r } 1 , \mathrm { r } 2$ and B are the radii of circles inscribed in the quadrilaterals AFIE, BDIF and CEID respectively, prove that r1/ ( $\mathrm { r } - \mathrm { r } 1$ ) + உ / ( $\mathrm { r } - \mathrm { r } 2$ ) + r3 / ( $\mathrm { r } - \mathrm { r } 3$ ) = ( r 1 B B ) / ( $\mathrm { r } - \mathrm { r } 1$ ) ( $\mathrm { r } - \mathrm { r } 2$ ) ( $\mathrm { r } - \mathrm { r } 3$ )
3. (a) In any triangle ABC , prove that
$$\operatorname { Cot } \mathrm { A } / \mathrm { b } + \cot \mathrm { B } / 2 + \cot \mathrm { C } / 2 = \mathrm { A } / 2 \cot \mathrm {~B} / 2 \cot \mathrm { C } / 2$$
(b) Let ABC be a triangle with incentre I and inradius r . Let $\mathrm { D } , \mathrm { E } , \mathrm { F }$ be the feet of the perpendiculars from I to the sides $\mathrm { BC } , \mathrm { CA }$ and AB respectively. If $\mathrm { r } 1 , \mathrm { r } 2$ and B are the radii of circles inscribed in the quadrilaterals AFIE, BDIF and CEID respectively, prove that\\
r1/ ( $\mathrm { r } - \mathrm { r } 1$ ) + உ / ( $\mathrm { r } - \mathrm { r } 2$ ) + r3 / ( $\mathrm { r } - \mathrm { r } 3$ ) = ( r 1 B B ) / ( $\mathrm { r } - \mathrm { r } 1$ ) ( $\mathrm { r } - \mathrm { r } 2$ ) ( $\mathrm { r } - \mathrm { r } 3$ )\\