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Question Number 51884 by ajfour last updated on 31/Dec/18

Commented by ajfour last updated on 31/Dec/18

$R e l a t e r a d i u s r o f t h e t w o c i r c l e s$ $(t h e s a m e) w i t h s i d e l e n g t h a o f$ $t h e e q u i l a t e r a l t r i a n g l e .$
Commented by mr W last updated on 01/Jan/19

$\frac{r}{a} = \frac{\sqrt{3} - \sqrt{2}}{2}$
Commented by ajfour last updated on 01/Jan/19

$A D = \frac{a \sqrt{3}}{2} = A S + S M + M Q + Q D$ $\frac{a \sqrt{3}}{2} = 2 r + \frac{r}{\cos 2 θ} + \frac{r}{\cos 2 θ} + r$ $\tan θ = \frac{2 r}{a} = t$ $\frac{a \sqrt{3}}{2} = 3 r + \frac{2 r [1 + {(\frac{2 r}{a})}^{2}]}{[1 - {(\frac{2 r}{a})}^{2}]}$ $l e t$ $\sqrt{3} = 3 t + \frac{2 t (1 + t^{2})}{1 - t^{2}}$ $\sqrt{3} - \sqrt{3} t^{2} = 3 t - 3 t^{3} + 2 t + 2 t^{3}$ $t^{3} - \sqrt{3} t^{2} - 5 t + \sqrt{3} = 0$
Commented by ajfour last updated on 01/Jan/19

	Answered by mr W last updated on 01/Jan/19

	$l e t α = ∠ P B C$ $B O = C O = \frac{a}{2 \cos α}$ $D O = \frac{a \tan α}{2}$ $Δ_{B O C} = \frac{a}{2} \times \frac{a \tan α}{2} = \frac{r}{2} (a + 2 \times \frac{a}{2 \cos α})$ $\frac{a \tan α}{2} = r (1 + \frac{1}{\cos α})$ $\Rightarrow r = \frac{a \sin α}{2 (1 + \cos α)} = \frac{a}{2} \tan \frac{α}{2} . . . (i)$ $B R = \frac{a \sin α}{\sin (\frac{π}{3} + α)}$ $A R = a - \frac{a \sin α}{\sin (\frac{π}{3} + α)} = \frac{\sin (\frac{π}{3} + α) - \sin α}{\sin (\frac{π}{3} + α)} \times a$ $= \frac{\sqrt{3} \cos α - \sin α}{\sqrt{3} \cos α + \sin α} \times a$ $C R = \frac{a \sin \frac{π}{3}}{\sin (\frac{π}{3} + α)} = \frac{\sqrt{3} a}{\sqrt{3} \cos α + \sin α}$ $Δ_{A R C} = \frac{a \sin \frac{π}{3}}{2} \times \frac{\sqrt{3} \cos α - \sin α}{\sqrt{3} \cos α + \sin α} \times a = \frac{r}{2} (a + \frac{\sqrt{3} \cos α - \sin α}{\sqrt{3} \cos α + \sin α} \times a + \frac{\sqrt{3} a}{\sqrt{3} \cos α + \sin α})$ $\Rightarrow r = \frac{\sqrt{3} \cos α - \sin α}{2 (1 + 2 \cos α)} \times a . . . (i i)$ $\Rightarrow \frac{\sin α}{1 + \cos α} = \frac{\sqrt{3} \cos α - \sin α}{1 + 2 \cos α}$ $\Rightarrow 2 \tan α + 3 \sin α = \sqrt{3} (1 + \cos α)$ $w i t h t = \tan \frac{α}{2}$ $\Rightarrow t^{3} - \sqrt{3} t^{2} - 5 t + \sqrt{3} = 0$ $\Rightarrow t = \sqrt{3} - \sqrt{2}$ $\Rightarrow \frac{r}{a} = \frac{t}{2} = \frac{\sqrt{3} - \sqrt{2}}{2} \approx 0.1589$
	Commented by ajfour last updated on 01/Jan/19

	$T h a n k y o u S i r .$
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