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Question Number 64289 by ajfour last updated on 16/Jul/19

Commented by ajfour last updated on 16/Jul/19

$F i n d s i n t e r m s o f r a d i i a a n d b .$
Commented by behi83417@gmail.com last updated on 16/Jul/19

$.$
	Answered by mr W last updated on 16/Jul/19

	$\cos α = \frac{s}{2 a}$ $\cos β = \frac{s}{2 b}$ $α + β + \frac{π}{3} = π$ $α = \frac{2 π}{3} - β$ $\cos α = \cos (\frac{2 π}{3} - β) = - \frac{1}{2} \cos β + \frac{\sqrt{3}}{2} \sin β$ $\frac{s}{2 a} = - \frac{s}{4 b} + \frac{\sqrt{3}}{2} \times \frac{\sqrt{4 b^{2} - s^{2}}}{2 b}$ $\frac{(2 b + a) s}{a} = \sqrt{3 (4 b^{2} - s^{2})}$ $\frac{b^{2} + a b + a^{2}}{a^{2}} s^{2} = 3 b^{2}$ $\Rightarrow s = \frac{\sqrt{3} a b}{\sqrt{a^{2} + a b + b^{2}}}$
	Commented by ajfour last updated on 17/Jul/19

	$t h a n k y o u s i r .$
	Commented by Tawa1 last updated on 17/Jul/19

	$God bless you sir . I want to study the solution$
	Answered by mr W last updated on 17/Jul/19

	Commented by mr W last updated on 17/Jul/19

	$\frac{E C}{A C} = \frac{A C}{D C}$ $\Rightarrow E C = \frac{s^{2}}{2 a}$ ${A E}^{2} = {A C}^{2} - {E C}^{2} = s^{2} - \frac{s^{4}}{4 a^{2}}$ $s i m i l a r l y$ $C F = \frac{s^{2}}{2 b}$ ${B F}^{2} = s^{2} - \frac{s^{4}}{4 b^{2}}$ ${A B}^{2} = {(E C + C F)}^{2} + {(B F - A E)}^{2}$ $s^{2} = {(\frac{s^{2}}{2 a} + \frac{s^{2}}{2 b})}^{2} + s^{2} - \frac{s^{4}}{4 a^{2}} + s^{2} - \frac{s^{4}}{4 b^{2}} - 2 \sqrt{(s^{2} - \frac{s^{4}}{4 a^{2}}) (s^{2} - \frac{s^{4}}{4 b^{2}})}$ $\frac{s^{4}}{2 a b} + s^{2} = 2 \sqrt{(s^{2} - \frac{s^{4}}{4 a^{2}}) (s^{2} - \frac{s^{4}}{4 b^{2}})}$ $\frac{s^{2}}{2 a b} + 1 = 2 \sqrt{(1 - \frac{s^{2}}{4 a^{2}}) (1 - \frac{s^{2}}{4 b^{2}})}$ ${(\frac{s^{2}}{2 a b} + 1)}^{2} = 4 (1 - \frac{s^{2}}{4 a^{2}}) (1 - \frac{s^{2}}{4 b^{2}})$ $\frac{s^{4}}{4 a^{2} b^{2}} + 1 + \frac{s^{2}}{a b} = 4 (1 - \frac{s^{2}}{4 a^{2}} - \frac{s^{2}}{4 b^{2}} + \frac{s^{4}}{16 a^{2} b^{2}})$ $\frac{s^{2}}{a b} = 3 - \frac{s^{2}}{a^{2}} - \frac{s^{2}}{b^{2}}$ $(\frac{1}{a^{2}} + \frac{1}{a b} + \frac{1}{b^{2}}) s^{2} = 3$ $(a^{2} + a b + b^{2}) s^{2} = 3 a^{2} b^{2}$ $\Rightarrow s = \frac{\sqrt{3} a b}{\sqrt{a^{2} + a b + b^{2}}}$
	Commented by Tawa1 last updated on 18/Jul/19

	$I appreciate your effort sir . God bless you more$
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