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Question Number 30233 by ajfour last updated on 18/Feb/18

Commented by ajfour last updated on 18/Feb/18

$F i n d m a x i m u m a r e a o f q u a d r i l a t e r a l$ $A B C D i f s i d e s A B, B C, C D, D A$ $b e a, b, c, d r e s p e c t i v e l y, w i t h$ $a < b < c < d .$
	Answered by mrW2 last updated on 19/Feb/18

	$l e t ∠ B = α, ∠ D = β < α$ $a^{2} + b^{2} - 2 a b \cos α = c^{2} + d^{2} - 2 c d \cos β$ $a b \sin α = c d \sin β \frac{d β}{d α}$ $A = \frac{1}{2} (a b \sin α + c d \sin β)$ $\frac{d A}{d α} = \frac{1}{2} [a b \cos α + c d \cos β (\frac{a b \sin α}{c d \sin β})]$ $\frac{d A}{d α} = \frac{1}{2} a b (\cos α + \frac{\sin α}{\tan β}) = 0$ $\Rightarrow \tan α = - \tan β$ $\Rightarrow α = π - β o r α + β = π \Rightarrow c y c l i c!$ $a^{2} + b^{2} + 2 a b \cos β = c^{2} + d^{2} - 2 c d \cos β$ $2 (a b + c d) \cos β = c^{2} + d^{2} - a^{2} - b^{2}$ $\Rightarrow \cos β = \frac{c^{2} + d^{2} - a^{2} - b^{2}}{2 (a b + c d)}$ $\Rightarrow \sin β = \sqrt{1 - \frac{{(c^{2} + d^{2} - a^{2} - b^{2})}^{2}}{4 {(a b + c d)}^{2}}} = \frac{\sqrt{(2 a b + 2 c d + c^{2} + d^{2} - a^{2} - b^{2}) (2 a b + 2 c d - c^{2} - d^{2} + a^{2} + b^{2})}}{2 (a b + c d)}$ $\Rightarrow \sin β = \frac{\sqrt{[{(c + d)}^{2} - {(a - b)}^{2}] [{(a + b)}^{2} - {(c - d)}^{2}]}}{2 (a b + c d)}$ $\Rightarrow \sin β = \frac{\sqrt{(- a + b + c + d) (a - b + c + d) (a + b - c + d) (a + b + c - d)}}{2 (a b + c d)}$ $\Rightarrow \sin α = \sin β$ $A_{m a x} = \frac{1}{2} (a b + c d) \sin β = \frac{1}{2} (a b + c d) \times \frac{\sqrt{(- a + b + c + d) (a - b + c + d) (a + b - c + d) (a + b + c - d)}}{2 (a b + c d)}$ $\Rightarrow A_{m a x} = \frac{1}{4} \sqrt{(- a + b + c + d) (a - b + c + d) (a + b - c + d) (a + b + c - d)}$ $w i t h s = \frac{a + b + c + d}{2}$ $\Rightarrow A_{m a x} = \sqrt{(s - a) (s - b) (s - c) (s - d)}$
	Commented by ajfour last updated on 19/Feb/18

	$W o n d e r f u l a n s w e r . T h a n k y o u S i r .$
	Commented by mrW2 last updated on 19/Feb/18

	$i f p e r i m e t e r o f q u a d r i l a t e r a l i s l, f i n d$ $t h e m a x i m a l p o s s i b l e a r e a o f i t .$ $l e t t h e f i r s t t h r e e s i d e l e n g t h s a r e x, y, z .$ $A = \frac{1}{4} \sqrt{(l - 2 x) (l - 2 y) (l - 2 z) (2 x + 2 y + 2 z - l)}$ $f (x, y, z) = \sqrt{(l - 2 x) (l - 2 y) (l - 2 z) (2 x + 2 y + 2 z - l)}$ $\frac{\partial f}{\partial x} = \frac{- 2 (2 x + 2 y + 2 z - l) + 2 (l - 2 x)}{2 f} = \frac{2 (l - 2 x - y - z)}{f} = 0$ $\Rightarrow l = 2 x + y + z$ $s i m i l a r l y$ $\Rightarrow l = 2 y + x + z$ $\Rightarrow l = 2 z + y + x$ $\Rightarrow 3 l = 4 (x + y + z)$ $\Rightarrow x + y + z = \frac{3 l}{4}$ $\Rightarrow l = 2 x + y + z = x + \frac{3 l}{4}$ $\Rightarrow x = \frac{l}{4}$ $s i m i l a r l y$ $\Rightarrow y = \frac{l}{4}$ $\Rightarrow z = \frac{l}{4}$ $\Rightarrow a r e a i s m a x i m a l i f {i t}^{'} s a s q u a r e .$ $A_{m a x} = \frac{l^{2}}{16}$
	Commented by ajfour last updated on 19/Feb/18

	$G o o d w a y t o v e r i f y S i r!$
	Commented by Tawa11 last updated on 06/Sep/24

	$Ohh, Great$
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