If $$a\,{\cos ^2}\left( {{C \over 2}} \right) + c\,{\cos ^2}\left( {{A \over 2}} \right) = {{3b} \over 2}$$

$$a\left[ {\cos C + 1} \right] + c\left[ {\cos A + 1} \right] = 3b$$

$$\left( {a + c} \right) + \left( {a\cos C + c\cos \,B} \right) = 3b$$

$$a + c + b = 3b$$ or $$a + c = 2b$$

or $$a,b,c$$ are in $$A.P.$$

$$AP = {2 \over 3}AD = {8 \over 3};\,\,PD = {4 \over 3};\,\,$$

Let $$PB=x$$

$$\tan {60^ \circ } = {{8/3} \over x}$$

or $$x = {8 \over {3\sqrt 3 }}$$

Area of $$\Delta ABD$$

$$ = {1 \over 2} \times 4 \times {8 \over {3\sqrt 3 }} = {{16} \over {3\sqrt 3 }}$$

$$\therefore$$ Area of $$\Delta ABC$$

$$ = 2 \times {{16} \over {3\sqrt 3 }} = {{32} \over {3\sqrt 3 }}$$

$$\left[ \, \right.$$ As median of a $$\Delta $$ divides it into two $$\Delta 's$$ of equal area. $$\left. \, \right]$$

$$\tan \left( {{\pi \over n}} \right) = {a \over {2r}};\,\,\sin \left( {{\pi \over n}} \right) = {a \over {2R}}$$

$$r + R = {a \over 2}\left[ {\cot {\pi \over n} + \cos ec{\pi \over n}} \right]$$


$$ = {a \over 2}\left[ {{{\cos {\pi \over n} + 1} \over {\sin {\pi \over n}}}} \right]$$

$$ = {a \over 2}\left[ {{{2{{\cos }^2}{\pi \over {2n}}} \over {2\sin {\pi \over {2n}}\cos {\pi \over {2n}}}}} \right]$$

$$ = {a \over 2}\cot {\pi \over {2\pi }}$$

$${r_1} > {r_2} > {r_3}$$

$$ \Rightarrow {\Delta \over {s - a}} > {\Delta \over {s - b}} > {\Delta \over {s - c}};$$

$$ \Rightarrow s - a < s - b < s - c$$

$$ \Rightarrow - a < - b < - c$$

$$ \Rightarrow a > b > c$$