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Question Number 187146 by Rupesh123 last updated on 14/Feb/23

	Answered by a.lgnaoui last updated on 14/Feb/23

	$△ S M I e t R N I s e m b l a b l e s$ $I c e n t r e d e S R M N S K = \frac{P Q}{2} = 3$ $△ B R Q R I O S e m b l a b l e s$ $\frac{O R}{O I} = \frac{\frac{R N}{2}}{H Q} = \frac{R Q}{B Q} \Rightarrow \frac{R N}{2 H Q} = \frac{R Q}{B Q}$ $H Q = 3 R Q = 6 B Q = B P + 6$ $\frac{R N}{6} = \frac{6}{6 + B P} (1)$ $△ A S K A B H s e m b l a b l e s$ $\cos \tan 60 = \frac{A K}{S K} = \frac{A H}{B H} \Rightarrow$ $\frac{A K}{3} = \frac{A K + 6}{B P + 3} = \sqrt{3} \Rightarrow A K = 3 \sqrt{3}$ $A l o r s \frac{3 \sqrt{3} + 6}{B P + 3} = \sqrt{3}$ $B P = \frac{3 \sqrt{3} + 6}{\sqrt{3}} - 3 = B P = 2 \sqrt{3}$ $(1) \Rightarrow R N = \frac{36}{6 + B P} = \frac{36}{6 + 2 \sqrt{3}} =$ $R N = 9 - 3 \sqrt{3}$ $F i n a l e m e n t : A i r e t o t a l e (A)$ $A = 2 \times (A i r e I R N)$ $A = 2 (\frac{R N \times O I}{2}) = R N \times O I = 27 - 9 \sqrt{3}$ $A i r e = 11, 41$
	Commented by Rupesh123 last updated on 14/Feb/23
	Excellent
	Answered by mr W last updated on 14/Feb/23

	$s a y s i d e l e n g t h o f e q u i l a t e r a l i s s .$ $\frac{\frac{\sqrt{3} s}{2} - 6}{\frac{\sqrt{3} s}{2}} = \frac{6}{s}$ $\Rightarrow s = 6 + 4 \sqrt{3}$ $B P = \frac{6}{\sqrt{3}} = 2 \sqrt{3}$ $B Q = 6 + 4 \sqrt{3} - 2 \sqrt{3} = 6 + 2 \sqrt{3}$ $\frac{T P}{R Q} = \frac{B P}{B Q}$ $\Rightarrow T P = \frac{2 \sqrt{3} \times 6}{6 + 2 \sqrt{3}} = 3 \sqrt{3} - 3$ $\Rightarrow S T = 6 - (3 \sqrt{3} - 3) = 3 (3 - \sqrt{3})$ $A_{b l u e} = \frac{6}{2} \times S T = 9 (3 - \sqrt{3}) \approx 11.412$
	Commented by Rupesh123 last updated on 14/Feb/23
	Excellent
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