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Question Number 48763 by mr W last updated on 28/Nov/18

Commented by mr W last updated on 28/Nov/18

$F i n d R i n t e r m s o f a, b, c .$
	Answered by ajfour last updated on 28/Nov/18

	Answered by ajfour last updated on 28/Nov/18

	$R \sin θ = \frac{a}{2}$ $2 R \cos (θ + α) = \sqrt{b^{2} + c^{2}}$ $\Rightarrow 2 R (\cos θ \cos α - \sin θ \sin α) = \sqrt{b^{2} + c^{2}}$ $2 R (\cos θ \times \frac{b}{\sqrt{b^{2} + c^{2}}} - \frac{a}{2 R} \times \frac{c}{\sqrt{b^{2} + c^{2}}}) = \sqrt{b^{2} + c^{2}}$ $\Rightarrow 2 b R \cos θ = b^{2} + c^{2} + a c$ $o r (4 R^{2} - a^{2}) b^{2} = {(b^{2} + c^{2} + a c)}^{2}$ $4 R^{2} = a^{2} + \frac{{(b^{2} + c^{2} + a c)}^{2}}{b^{2}}$ $R = \frac{1}{2} \sqrt{a^{2} + \frac{{(b^{2} + c^{2} + a c)}^{2}}{b^{2}}} .$
	Commented by mr W last updated on 28/Nov/18

	$c o r r e c t! t h a n k y o u s i r!$
	Commented by mr W last updated on 28/Nov/18

	$a l t e r n a t i v e w a y :$ $c i r c u m c i r c l e o f a t r i a n g l e w i t h s i d e s :$ $a$ $\sqrt{b^{2} + c^{2}}$ $\sqrt{b^{2} + {(a + c)}^{2}}$ $R = \frac{a \sqrt{(b^{2} + c^{2}) (b^{2} + {(a + c)}^{2})}}{4 A}$ $A = \frac{a b}{2}$ $R = \frac{a \sqrt{(b^{2} + c^{2}) (b^{2} + {(a + c)}^{2})}}{2 a b}$ $\Rightarrow R = \frac{\sqrt{(b^{2} + c^{2}) (b^{2} + {(a + c)}^{2})}}{2 b}$
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