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Question Number 47735 by behi83417@gmail.com last updated on 14/Nov/18

Commented by mr W last updated on 14/Nov/18

$w e o n l y n e e d t o k n o w t h e d i s t a n c e s t o$ $t h r e e v e r t e x e s . t h e d i s t a n c e t o t h e$ $f o u r t h v e r t e x i s n o t i n d e p e n d e n t .$
Commented by behi83417@gmail.com last updated on 14/Nov/18

$A B C D : s q u a r e .$ $A E = 2, B E = 4, C E = 6$ $. . . . . \to A B = ?$
Commented by MJS last updated on 14/Nov/18

$(1) x^{2} + y^{2} = 4$ $(2) {(x - a)}^{2} + y^{2} = 16$ $(3) {(x - a)}^{2} + {(y - b)}^{2} = 36$ $(4) x^{2} + {(y - b)}^{2} = 12$ $(2) - (1) - 2 a x + a^{2} = 12$ $(3) - (4) - 2 a x + a^{2} = 24$ $\Rightarrow {it}^{'} s also inconsistent for A B C D being a$ $rectangle$
Commented by MJS last updated on 14/Nov/18

$(1) x^{2} + y^{2} = 4$ $(2) {(x - a)}^{2} + y^{2} = 16$ $(3) {(x - a)}^{2} + {(y - a)}^{2} = 36$ $(2) - (1) - 2 a x + a^{2} = 12 \Rightarrow x = \frac{a^{2} - 12}{2 a}$ $(3) - (2) - 2 a y + a^{2} = 20 \Rightarrow y = \frac{a^{2} - 20}{2 a}$ $(1) {(a^{2} - 12)}^{2} + {(a^{2} - 20)}^{2} = 16 a^{2}$ $a^{4} - 40 a^{2} + 272 = 0$ $a_{1} = 2 \sqrt{5 - 2 \sqrt{2}} \approx 2.95 \Rightarrow E outside square$ $a_{2} = 2 \sqrt{5 + 2 \sqrt{2}} \approx 5.60 \Rightarrow E inside square$
Commented by behi83417@gmail.com last updated on 14/Nov/18

$t h a n k y o u s o m u c h s i r M J S . p e r f e c t .$
	Answered by mr W last updated on 14/Nov/18

	$a = d i s t a n c e t o v e r t e x A = 2$ $b = d i s t a n c e t o v e r t e x B = 4$ $c = d i s t a n c e t o v e r t e x C = 6$ $l = s i d e l e n g t h o f s q u a r e$ $x, y = d i s t a n c e t o s i d e B C a n d A B$ $x^{2} + y^{2} = b^{2}$ $a^{2} = {(l - x)}^{2} + y^{2} = l^{2} - 2 l x + x^{2} + y^{2} = l^{2} + b^{2} - 2 l x$ $\Rightarrow x = \frac{l^{2} + b^{2} - a^{2}}{2 l}$ $c^{2} = x^{2} + {(l - y)}^{2} = x^{2} + y^{2} + l^{2} - 2 l y = l^{2} + b^{2} - 2 l y$ $\Rightarrow y = \frac{l^{2} + b^{2} - c^{2}}{2 l}$ ${(\frac{l^{2} + b^{2} - a^{2}}{2 l})}^{2} + {(\frac{l^{2} + b^{2} - c^{2}}{2 l})}^{2} = b^{2}$ $2 l^{4} + 2 (2 b^{2} - a^{2} - c^{2}) l^{2} + [2 b^{4} + a^{4} + c^{4} - 2 (a^{2} + c^{2}) b^{2}] = 4 b^{2} l^{2}$ $2 l^{4} - 2 (a^{2} + c^{2}) l^{2} + [2 b^{4} + a^{4} + c^{4} - 2 (a^{2} + c^{2}) b^{2}] = 0$ $l^{2} = \frac{(a^{2} + c^{2}) \pm \sqrt{(a^{4} + c^{4} + 2 a^{2} c^{2}) - 2 [2 b^{4} + a^{4} + c^{4} - 2 (a^{2} + c^{2}) b^{2}]}}{2}$ $l^{2} = \frac{a^{2} + c^{2} \pm \sqrt{[{(a + c)}^{2} - 2 b^{2}] [2 b^{2} - {(a - c)}^{2}]}}{2}$ $l^{2} = \frac{a^{2} + c^{2} \pm \sqrt{(a + \sqrt{2} b + c) (- a + \sqrt{2} b + c) (a - \sqrt{2} b + c) (a + \sqrt{2} b - c)}}{2}$ $l^{2} = \frac{2^{2} + 6^{2} \pm \sqrt{(64 - 2 \times 16) (2 \times 16 - 16)}}{2}$ $l^{2} = 20 \pm 8 \sqrt{2}$ $\Rightarrow l = 2 \sqrt{5 \pm 2 \sqrt{2}} = 5.59 o r 2.95$
	Commented by behi83417@gmail.com last updated on 14/Nov/18

	$e x c e l l e n t s i r m r W . t h a n k y o u v e r y$ $m u c h .$
	Commented by mr W last updated on 14/Nov/18

	Commented by mr W last updated on 14/Nov/18

	$t h i s i s h o w t o d r a w t h e r e q u e s t e d s q u a r e .$ $d r a w t h r e e c i r c l e s w i t h r a d i i a, b, c f r o m$ $t h e s a m e c e n t e r P .$ $t a k e a f i x e d p o i n t A o n t h e f i r s t c i r c l e .$ $s e l e c t t h r e e p o i n t s o n t h e s e c o n d c i r c l e,$ $s a y s 1, s 2 a n d s 3 .$ $w i t h {A S}_{1} a s s i d e l e n g t h c o n s t r u c t a$ $s q u a r e w h o s e v e r t e x o p p o s i t e t o S_{1} i s T_{1} .$ $s i m i l a r l y w e g e t f r o m o t h e r t w o s q u a r e s$ $p o i n t s T_{2} a n d T_{3} .$ $d r a w a f o u r t h c i r c l e t h r o u g h p o i n t s$ $T_{1}, T_{2} a n d T_{3} .$ $t h i s f o u r t h c i r c l e i n t e r s e c t s t h e t h i r d$ $c i r c l e a t t w o p o i n t s, o r a t o n e p o i n t, o r$ $n o t a t a l l .$ $l e t s s a y t h e i n t e r s e c t i o n p o i n t s a r e$ $B a n d B^{'} .$ $A B o r {A B}^{'} i s t h e s i d e l e n g t h o f t h e$ $r e q u e s t e d s q u a r e .$
	Commented by behi83417@gmail.com last updated on 14/Nov/18

	$i n t r e s t i n g! y o u a r e a m a z i n g m a s t e r .$ $t h a n k y o u v e r y m u c h d e a r .$
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