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Question Number 198421 by mr W last updated on 19/Oct/23

Commented by mr W last updated on 19/Oct/23

$g i v e n t h a t t h e m i d p o i n t s o f t h e s i d e s$ $o f a c o n v e x q u a d r i l a t e r i a l a r e$ $c o n c y c l i c . i f t h e l e n g t h e s o f t h r e e o f$ $i t s s i d e s a r e a, b, c r e s p e c t i v e l y, f i n d$ $t h e l e n g t h o f i t s f o u r t h s i d e .$
Commented by som(math1967) last updated on 20/Oct/23

$M i d p t f o r m a r e c t a n g l e (c y c l i c p a r e l a l l o g r a m)$ $s o A B = A D . a m i c o r r e c t s i r ?$
Commented by mr W last updated on 20/Oct/23

$y e s$
	Answered by mr W last updated on 20/Oct/23

	Commented by mr W last updated on 20/Oct/23

	$E H = F G = \frac{B D}{2}$ $E H / / F G / / B D$ $E F = H G = \frac{A C}{2}$ $E F / / H G / / A C$ $\Rightarrow E F G H i s p a r a l l e l o g r a m$ $s i n c e E F G H a r e c o n c y c l i c$ $\Rightarrow E F G H i s r e c t a n g l e$ $\Rightarrow A C ⊥ B D$ $a^{2} + c^{2} = {A K}^{2} + {B K}^{2} + {C K}^{2} + {D K}^{2}$ $b^{2} + d^{2} = {C K}^{2} + {B K}^{2} + {A K}^{2} + {D K}^{2}$ $\Rightarrow a^{2} + c^{2} = b^{2} + d^{2}$ $\Rightarrow d = \sqrt{a^{2} + c^{2} - b^{2}}$
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