show that (((a+b+c)^2 )/(a^2 +b^2 +c^2 ))= ((cot (1/2)A+cot (1/2)B+cot (1/2)C)/(cot A+cot B+cot C)) please help


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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
	$We have cot (A/2)=(√((s(s−a))/((s−b)(s−c)))) −−(1) Let ((sin A)/a)=((sin B)/b)=((sin C)/c)=k ⇒cot A=((cos A)/(sin A))=(((b^2 +c^2 −a^2 )/(2bc))/(ak))=((b^2 +c^2 −a^2 )/(2abck)) −−(2) Using (1) & (2) R.H.S.=((cot (1/2)A+cot (1/2)B+cot (1/2)C)/(cot A+cot B+cot C)) =(((√((s(s−a))/((s−b)(s−c))))+(√((s(s−b))/((s−a)(s−c))))+(√((s(s−c))/((s−b)(s−a)))))/(((b^2 +c^2 −a^2 )/(2abck))+((a^2 +c^2 −b^2 )/(2abck))+((a^2 +b^2 −c^2 )/(2abck)))) =((2abck)/(√((s−a)(s−b)(s−c)))).(((√(s(s−a)^2 ))+(√(s(s−b)^2 ))+(√(s(s−c)^2 )))/(b^2 +c^2 −a^2 +a^2 +c^2 −b^2 +a^2 +b^2 −c^2 )) =((2abck)/(√(s(s−a)(s−b)(s−c)))).((s{(s−a)+(s−b)+(s−c)})/(a^2 +b^2 +c^2 )) =((2abck)/((1/2)bcsin A)).((s{3s−(a+b+c)})/(a^2 +b^2 +c^2 )) =4.((s{3s−2s})/(a^2 +b^2 +c^2 )) =4.(s^2 /(a^2 +b^2 +c^2 )) =(((2s)^2 )/(a^2 +b^2 +c^2 )) =(((a+b+c)^2 )/(a^2 +b^2 +c^2 ))=L.H.S.$
	$W e h a v e$ $\cot \frac{A}{2} = \sqrt{\frac{s (s - a)}{(s - b) (s - c)}} - - (1)$ $L e t \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} = k$ $\Rightarrow \cot A = \frac{\cos A}{\sin A} = \frac{\frac{b^{2} + c^{2} - a^{2}}{2 b c}}{a k} = \frac{b^{2} + c^{2} - a^{2}}{2 a b c k} - - (2)$ $U s i n g (1) & (2)$ $R . H . S . = \frac{\cot \frac{1}{2} A + \cot \frac{1}{2} B + \cot \frac{1}{2} C}{\cot A + \cot B + \cot C}$ $= \frac{\sqrt{\frac{s (s - a)}{(s - b) (s - c)}} + \sqrt{\frac{s (s - b)}{(s - a) (s - c)}} + \sqrt{\frac{s (s - c)}{(s - b) (s - a)}}}{\frac{b^{2} + c^{2} - a^{2}}{2 a b c k} + \frac{a^{2} + c^{2} - b^{2}}{2 a b c k} + \frac{a^{2} + b^{2} - c^{2}}{2 a b c k}}$ $= \frac{2 a b c k}{\sqrt{(s - a) (s - b) (s - c)}} . \frac{\sqrt{s {(s - a)}^{2}} + \sqrt{s {(s - b)}^{2}} + \sqrt{s {(s - c)}^{2}}}{b^{2} + c^{2} - a^{2} + a^{2} + c^{2} - b^{2} + a^{2} + b^{2} - c^{2}}$ $= \frac{2 a b c k}{\sqrt{s (s - a) (s - b) (s - c)}} . \frac{s {(s - a) + (s - b) + (s - c)}}{a^{2} + b^{2} + c^{2}}$ $= \frac{2 a b c k}{\frac{1}{2} b c \sin A} . \frac{s {3 s - (a + b + c)}}{a^{2} + b^{2} + c^{2}}$ $= 4 . \frac{s {3 s - 2 s}}{a^{2} + b^{2} + c^{2}}$ $= 4 . \frac{s^{2}}{a^{2} + b^{2} + c^{2}}$ $= \frac{{(2 s)}^{2}}{a^{2} + b^{2} + c^{2}}$ $= \frac{{(a + b + c)}^{2}}{a^{2} + b^{2} + c^{2}} = L . H . S .$
	Commented by NECx last updated on 20/Oct/17

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