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Question Number 22536 by NECx last updated on 20/Oct/17

s h o w t h a t \frac{{(a + b + c)}^{2}}{a^{2} + b^{2} + c^{2}} =

\frac{\cot \frac{1}{2} A + \cot \frac{1}{2} B + \cot \frac{1}{2} C}{\cot A + \cot B + \cot C}

p l e a s e h e l p

Commented by ajfour last updated on 20/Oct/17

A n o t h e r s o l u t i o n : s e e Q . 22576

Answered by $@ty@m last updated on 20/Oct/17

W e h a v e

\cot \frac{A}{2} = \sqrt{\frac{s (s - a)}{(s - b) (s - c)}} - - (1)

L e t \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} = k

\Rightarrow \cot A = \frac{\cos A}{\sin A} = \frac{\frac{b^{2} + c^{2} - a^{2}}{2 b c}}{a k} = \frac{b^{2} + c^{2} - a^{2}}{2 a b c k} - - (2)

U s i n g (1) & (2)

R . H . S . = \frac{\cot \frac{1}{2} A + \cot \frac{1}{2} B + \cot \frac{1}{2} C}{\cot A + \cot B + \cot C}

= \frac{\sqrt{\frac{s (s - a)}{(s - b) (s - c)}} + \sqrt{\frac{s (s - b)}{(s - a) (s - c)}} + \sqrt{\frac{s (s - c)}{(s - b) (s - a)}}}{\frac{b^{2} + c^{2} - a^{2}}{2 a b c k} + \frac{a^{2} + c^{2} - b^{2}}{2 a b c k} + \frac{a^{2} + b^{2} - c^{2}}{2 a b c k}}

= \frac{2 a b c k}{\sqrt{(s - a) (s - b) (s - c)}} . \frac{\sqrt{s {(s - a)}^{2}} + \sqrt{s {(s - b)}^{2}} + \sqrt{s {(s - c)}^{2}}}{b^{2} + c^{2} - a^{2} + a^{2} + c^{2} - b^{2} + a^{2} + b^{2} - c^{2}}

= \frac{2 a b c k}{\sqrt{s (s - a) (s - b) (s - c)}} . \frac{s {(s - a) + (s - b) + (s - c)}}{a^{2} + b^{2} + c^{2}}

= \frac{2 a b c k}{\frac{1}{2} b c \sin A} . \frac{s {3 s - (a + b + c)}}{a^{2} + b^{2} + c^{2}}

= 4 . \frac{s {3 s - 2 s}}{a^{2} + b^{2} + c^{2}}

= 4 . \frac{s^{2}}{a^{2} + b^{2} + c^{2}}

= \frac{{(2 s)}^{2}}{a^{2} + b^{2} + c^{2}}

= \frac{{(a + b + c)}^{2}}{a^{2} + b^{2} + c^{2}} = L . H . S .

Commented by NECx last updated on 20/Oct/17

w o w \dots . . i^{'} m m o s t g r a t e f u l .

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