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Question Number 168550 by Shirnivak last updated on 13/Apr/22

	Answered by mr W last updated on 13/Apr/22

	Commented by mr W last updated on 13/Apr/22

	$R = r a d i u s o f c i r c u m c i r c l e o f Δ A B C$ $O A = O B = \frac{c}{\sqrt{2}}$ $A G = A C = b$ $B D = B C = a$ $A F = \sqrt{2} b$ $B E = \sqrt{2} a$ ${O G}^{2} = {(\frac{c}{\sqrt{2}})}^{2} + b^{2} - 2 \times \frac{c}{\sqrt{2}} \times b \cos (A + \frac{π}{2} + \frac{π}{4})$ ${O G}^{2} = \frac{c^{2}}{2} + b^{2} + \sqrt{2} c b \sin (A + \frac{π}{4})$ ${O G}^{2} = \frac{c^{2}}{2} + b^{2} + c b (\sin A + \cos A)$ ${O G}^{2} = \frac{c^{2}}{2} + b^{2} + c b (\frac{a}{2 R} + \frac{c^{2} + b^{2} - a^{2}}{2 c b})$ ${O G}^{2} = c^{2} + \frac{a b c}{2 R} + \frac{3 b^{2} - a^{2}}{2}$ ${O E}^{2} = {(\frac{c}{\sqrt{2}})}^{2} + {(\sqrt{2} a)}^{2} - 2 c a \cos (B + \frac{π}{4} + \frac{π}{4})$ ${O E}^{2} = \frac{c^{2}}{2} + 2 a^{2} + 2 c a \sin B$ ${O E}^{2} = \frac{c^{2}}{2} + 2 a^{2} + \frac{a b c}{R}$ ${O G}^{2} + {O E}^{2} = \frac{3}{2} (a^{2} + b^{2} + c^{2} + \frac{a b c}{R})$ $s i m i l a r l y$ ${O D}^{2} + {O F}^{2} = \frac{3}{2} (a^{2} + b^{2} + c^{2} + \frac{a b c}{R})$ $\Rightarrow {O G}^{2} + {O E}^{2} = {O D}^{2} + {O F}^{2}$
	Commented by Tawa11 last updated on 13/Apr/22

	$Great sir .$
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