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Question Number 54334 by ajfour last updated on 02/Feb/19

Commented by ajfour last updated on 02/Feb/19

$S i r d^{2} \neq a^{2} + b^{2} (c o z d {i s n}^{'} t d i a g o n a l)$
	Answered by mr W last updated on 02/Feb/19

	${(a - 2 r)}^{2} + {(b - 2 r)}^{2} = d^{2}$ $\Rightarrow a^{2} + b^{2} - 4 r (a + b) = d^{2} - 8 r^{2}$ $\sqrt{a^{2} + b^{2}} = (a + b) - 2 r$ $\Rightarrow a^{2} + b^{2} = {(a + b)}^{2} - 4 r (a + b) + 4 r^{2}$ $l e t X = a + b, Y = a^{2} + b^{2}$ $\Rightarrow Y - 4 r X = d^{2} - 8 r^{2}$ $\Rightarrow Y = X^{2} - 4 r X + 4 r^{2}$ $\Rightarrow d^{2} - 8 r^{2} + 4 r X = X^{2} - 4 r X + 4 r^{2}$ $\Rightarrow X^{2} - 8 r X + 12 r^{2} - d^{2} = 0$ $\Rightarrow X = a + b = 4 r + \sqrt{4 r^{2} + d^{2}}$ $\Rightarrow Y = a^{2} + b^{2} = d^{2} + 8 r^{2} + 4 r \sqrt{4 r^{2} + d^{2}}$ $a - b = \sqrt{2 (a^{2} + b^{2}) - {(a + b)}^{2}}$ $a - b = \sqrt{2 (d^{2} + 8 r^{2} + 4 r \sqrt{4 r^{2} + d^{2}}) - {(4 r + \sqrt{4 r^{2} + d^{2}})}^{2}}$ $\Rightarrow a - b = \sqrt{d^{2} - 4 r^{2}}$ $\Rightarrow a = 2 r + \sqrt{r^{2} + {(\frac{d}{2})}^{2}} + \sqrt{{(\frac{d}{2})}^{2} - r^{2}}$ $\Rightarrow b = 2 r + \sqrt{r^{2} + {(\frac{d}{2})}^{2}} - \sqrt{{(\frac{d}{2})}^{2} - r^{2}}$
	Commented by ajfour last updated on 02/Feb/19

	$T h a n k y o u S i r, i h o p e y o u l i k e d i t .$ $I l i k e d i t m y s e l f .$
	Commented by mr W last updated on 02/Feb/19

	$l i k e a l l o t h e r s o f y o u r s, t h i s i s a l s o a$ $n i c e a n d i n t e r e s t i n g q u e s t i o n s i r!$
	Commented by ajfour last updated on 02/Feb/19

	$T h a n k s f o r t h e a p p r e c i a t i o n, S i r,$ $t h a t s h o w i l e a r n f r o m y o u .$
	Commented by Rasheed.Sindhi last updated on 03/Feb/19

	$S i r M r W, I {d i d n}^{'} t u n d e r s t a n d t h e$ $3 r d l i n e :$ $\sqrt{a^{2} + b^{2}} = (a + b) - 2 r$ $P l e x p l a i n .$
	Commented by mr W last updated on 03/Feb/19

	Commented by mr W last updated on 03/Feb/19

	$d i a g o n a l = \sqrt{a^{2} + b^{2}} = (a - r) + (b - r) = (a + b) - 2 r$
	Commented by Rasheed.Sindhi last updated on 03/Feb/19

	$L o T o F T h a n K s S i R!$
	Commented by mr W last updated on 03/Feb/19

	$t h a n k s f o r r e v i e w i n g s i r!$
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