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Question Number 49433 by behi83417@gmail.com last updated on 06/Dec/18

Commented by ajfour last updated on 06/Dec/18

$c i r c l e r a d i u s, S i r ?$
Commented by behi83417@gmail.com last updated on 06/Dec/18

$B C = x, D C = y, c i r c l e r a d i u s = r .$ $f i n d B D, i n t e r m s o f : x, y, r .$
Commented by behi83417@gmail.com last updated on 07/Dec/18

$B D = a$ $∡ B A D = α = B \overset{⌢}{C D}$ $∡ B C D = β = \frac{B \overset{⌢}{D}}{2} = \frac{360 - B \overset{⌢}{C D}}{2} = 180 - \frac{B \overset{⌢}{C D}}{2}$ $c o s B \overset{}{C D} = - c o s (\frac{1}{2} B \overset{}{A D})$ ${c o s}^{2} β = \frac{1 + c o s α}{2} \Rightarrow 2 {c o s}^{2} β - c o s α = 1$ $2 {(\frac{x^{2} + y^{2} - a^{2}}{2 x y})}^{2} - \frac{2 r^{2} - a^{2}}{2 r^{2}} = 1 \Rightarrow$ $4 r^{2} {(x^{2} + y^{2} - a^{2})}^{2} - 4 x^{2} y^{2} (2 r^{2} - a^{2}) = 8 r^{2} x^{2} y^{2}$ $4 r^{2} a^{4} - 4 a^{2} [2 r^{2} (x^{2} + y^{2}) - x^{2} y^{2}] + 4 r^{2} {(x^{2} + y^{2})}^{2} - 16 x^{2} y^{2} r^{2} = 0$
Commented by behi83417@gmail.com last updated on 07/Dec/18

$y o u a r e r i g h t s i r . t h i s i s m y t y p o . n o w$ $i t i s f i x e d . t h a n k s a l o t f o r a t t e n t i o n .$
Commented by mr W last updated on 07/Dec/18

$s i r, b u t y o u r e q n . d i f f e r s f r o m m i n e .$ $y o u r s i s a f t e r r e a r r a n g e m e n t$ $a^{4} - [(x^{2} + y^{2}) - \frac{x^{2} y^{2}}{r^{2}}] a^{2} + {(x^{2} - y^{2})}^{2} = 0$ $m i n e$ $a^{4} - [2 (x^{2} + y^{2}) - \frac{x^{2} y^{2}}{r^{2}}] a^{2} + {(x^{2} - y^{2})}^{2} = 0$ $c a n y o u p l e a s e c h e c k ?$
Commented by mr W last updated on 07/Dec/18

$t h a n k s f o r c h e c k i n g s i r!$
	Answered by mr W last updated on 06/Dec/18

	$l e t a = B D$ $R = \frac{x y a}{\sqrt{(x + y + a) (- x + y + a) (x - y + a) (x + y - a)}}$ $R = \frac{x y a}{\sqrt{[{(x + y)}^{2} - a^{2}] [a^{2} - {(x - y)}^{2}]}}$ $[{(x + y)}^{2} - a^{2}] [a^{2} - {(x - y)}^{2}] = {(\frac{x y}{R})}^{2} a^{2}$ $a^{4} - [2 (x^{2} + y^{2}) - {(\frac{x y}{R})}^{2}] a^{2} + {(x^{2} - y^{2})}^{2} = 0$ $a^{2} = \frac{2 (x^{2} + y^{2}) - {(\frac{x y}{R})}^{2} \pm \sqrt{{[2 (x^{2} + y^{2}) - {(\frac{x y}{R})}^{2}]}^{2} - 4 {(x^{2} - y^{2})}^{2}}}{2}$ $a^{2} = \frac{2 (x^{2} + y^{2}) - {(\frac{x y}{R})}^{2} \pm \sqrt{[4 x^{2} - {(\frac{x y}{R})}^{2}] [4 y^{2} - {(\frac{x y}{R})}^{2}]}}{2}$ $a^{2} = \frac{2 (x^{2} + y^{2}) - {(\frac{x y}{R})}^{2} \pm \sqrt{(2 x + \frac{x y}{R}) (2 x - \frac{x y}{R}) (2 y + \frac{x y}{R}) (2 y - \frac{x y}{R})}}{2}$ $\Rightarrow a = \sqrt{\frac{2 (x^{2} + y^{2}) - {(\frac{x y}{R})}^{2} \pm \sqrt{(2 x + \frac{x y}{R}) (2 x - \frac{x y}{R}) (2 y + \frac{x y}{R}) (2 y - \frac{x y}{R})}}{2}}$
	Commented by behi83417@gmail.com last updated on 06/Dec/18

	$w o w! p e r f e c t s i r . t h a n k s i n a d v a n c e .$
	Commented by mr W last updated on 07/Dec/18

	$c h e c k w i t h x = y = \sqrt{2} R$ $\Rightarrow a = \sqrt{\frac{2 (4 R^{2}) - {(2 R)}^{2} \pm \sqrt{{(2 \sqrt{2} R + 2 R)}^{2} {(2 \sqrt{2} R - 2 R)}^{2}}}{2}}$ $\Rightarrow a = \sqrt{\frac{4 R^{2} \pm 4 R^{2}}{2}} = 2 R o r 0 \Rightarrow c o r r e c t$
	Answered by behi83417@gmail.com last updated on 06/Dec/18

	Commented by behi83417@gmail.com last updated on 06/Dec/18

	$∡ F B C = ∡ F D C = 90^{∙}$ ${B F}^{2} = 4 r^{2} - x^{2}, {D F}^{2} = 4 r^{2} - y^{2}$ $b y u s i n g {petolemy}^{'} s t e o r e m i n : F D C B :$ $B D . C F = B C . D F + B F . C D$ $2 r . B D = x . \sqrt{4 r^{2} - y^{2}} + y . \sqrt{4 r^{2} - x^{2}}$ $\Rightarrow B D = \frac{x . \sqrt{4 r^{2} - y^{2}} + y . \sqrt{4 r^{2} - x^{2}}}{2 r} . ◼$
	Commented by behi83417@gmail.com last updated on 07/Dec/18

	$c h e c h f o r : x = y = r \sqrt{2}$ $B D = \frac{r \sqrt{2} . r \sqrt{2} + r \sqrt{2} . r \sqrt{2}}{2 r} = \frac{4 r^{2}}{2 r} = 2 r [o k] .$
	Answered by behi83417@gmail.com last updated on 07/Dec/18

	$∡ B A C = α, ∡ D A C = β$ $c o s α = \frac{2 r^{2} - x^{2}}{2 r^{2}}, s i n α = \frac{x}{2 r^{2}} \sqrt{4 r^{2} - x^{2}}$ $c o s β = \frac{2 r^{2} - y^{2}}{2 r^{2}}, s i n β = \frac{y}{2 r^{2}} \sqrt{4 r^{2} - y^{2}}$ $c o s (α + β) = \frac{(2 r^{2} - x^{2}) (2 r^{2} - y^{2})}{4 r^{4}} - \frac{x y}{4 r^{4}} \sqrt{(4 r^{2} - x^{2}) (4 r^{2} - y^{2})}$ ${B D}^{2} = 2 r^{2} - 2 r^{2} . c o s (α + β) =$ $= \frac{xy . \sqrt{(4 r^{2} - x^{2}) (4 r^{2} - y^{2})} - (2 r^{2} - x^{2}) (2 r^{2} - y^{2}) + 4 r^{4}}{2 r^{2}} .$ $= \frac{xy . \sqrt{(4 r^{2} - x^{2}) (4 r^{2} - y^{2})} + 2 r^{2} (x^{2} + y^{2}) - x^{2} y^{2}}{2 r^{2}} .$ $B D = \sqrt{\frac{x y}{2 r^{2}} . \sqrt{(4 r^{2} - x^{2}) (4 r^{2} - y^{2})} + (x^{2} + y^{2}) - {(\frac{x y}{r \sqrt{2}})}^{2}} .$
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