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Question Number 104468 by ajfour last updated on 21/Jul/20

Commented by ajfour last updated on 21/Jul/20

$F i n d r a n d h e n c e a r e a o f △ A B C i n$ $t e r m s o f R .$
Commented by mr W last updated on 22/Jul/20

${i t}^{'} s n o t c l e a r h o w t h e s m a l l c i r c l e i s$ $u n q u e l y d e f i n e d . t h e b l u e c i r c l e i n$ $t h e d i a g r a m i s a l s o v a l i d ?$
Commented by mr W last updated on 22/Jul/20

Commented by malwaan last updated on 22/Jul/20

$m r W$ $n o t e t h a t$ $∣ A B ∣=∣ A F ∣; ∣ B F ∣=∣ B E ∣$ $i n △ B D E : ∡ B E D = 90 °$ $; ∣ E D ∣= r; ∣ B D ∣= 2 R$ $\Rightarrow∣ B E ∣= \sqrt{4 R^{2} - r^{2}}$ $. . . .$
Commented by mr W last updated on 22/Jul/20

$t h e n r i s n o t u n i q u e, i t h i n k .$
Commented by ajfour last updated on 22/Jul/20

$l e t s t h e n h a v e R a n d r g i v e n,$ $S i r; t o f i n d a r e a o f △ A B C,$ $i n t e r m s o f R a n d r .$
Commented by malwaan last updated on 22/Jul/20

$i t d e p e n d o n t h e l o c a t i o n$ $o f t h e p o i n t A$ $S O y o u a r e a b s o l u t e l y r i g h t$ $m r W$
	Answered by mr W last updated on 22/Jul/20

	Commented by mr W last updated on 22/Jul/20

	$\sin β = \frac{r}{2 R}$ $\Rightarrow β = \sin^{- 1} \frac{r}{2 R}$ $\Rightarrow α = 2 β$ $A B = \frac{R}{\tan \frac{α}{2}} = \frac{R \sqrt{4 R^{2} - r^{2}}}{r}$ $β + ∠ C = \frac{π}{2} - α$ $∠ C = \frac{π}{2} - α - β = \frac{π}{2} - 3 β$ $\frac{B C}{\sin α} = \frac{A B}{\sin ∠ C} = \frac{A B}{\cos 3 β}$ $\Rightarrow B C = \frac{\sin 2 β A B}{\cos 3 β}$ $Δ_{A B C} = \frac{A B \times B C \times \sin (\frac{π}{2} + β)}{2}$ $= \frac{{A B}^{2} \times \sin 2 β \times \cos β}{2 \cos 3 β}$ $= \frac{R^{2} (4 R^{2} - r^{2}) \sin β \times \cos^{2} β}{r^{2} \cos β (4 \cos^{2} β - 3)}$ $= \frac{R^{2} (4 R^{2} - r^{2}) \frac{r}{2 R} \sqrt{1 - \frac{r^{2}}{4 R^{2}}}}{r^{2} (4 (1 - \frac{r^{2}}{4 R^{2}}) - 3)}$ $= \frac{R^{2} {(4 R^{2} - r^{2})}^{\frac{3}{2}}}{4 r (R^{2} - r^{2})}$
	Commented by ajfour last updated on 24/Jul/20

	$t h a n k s s i r, s h a l l r e v i e w i t, b i t$ $u n w e l l a g a i n . .$ $E l e g a n t S o l v i n g S i r, t o o g o o d t o$ $f o l l o w i t e v e n, t h a n k s a l o t!$
	Commented by mr W last updated on 23/Jul/20

	$g e t w e l l!$
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