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Question Number 218607 by Spillover last updated on 13/Apr/25

	Answered by mr W last updated on 13/Apr/25

	Commented by mr W last updated on 13/Apr/25

	$b i g g e r c i r c l e :$ ${(x - \frac{s}{2})}^{2} + {(y - \frac{s}{2})}^{2} = \frac{s^{2}}{2}$ $s m a l l e r c i r c l e :$ ${(x - \frac{s}{2})}^{2} + {(y - \frac{3 s}{8})}^{2} = {(\frac{5 s}{8})}^{2}$ ${(- \frac{s}{2})}^{2} + {(y_{G} - \frac{3 s}{8})}^{2} = {(\frac{5 s}{8})}^{2}$ $\Rightarrow y_{G} = \frac{3 s}{4}$ $e q n . o f G H :$ $y = s + \frac{1}{2} (x - \frac{s}{2}) = \frac{x}{2} + \frac{3 s}{4}$ ${(x - \frac{s}{2})}^{2} + {(\frac{x}{2} + \frac{3 s}{4} - \frac{s}{2})}^{2} = \frac{s^{2}}{2}$ $5 x^{2} - 3 s x - \frac{3 s^{2}}{4} = 0$ $x = (\frac{3 \pm 2 \sqrt{6}}{10}) s$ $\Rightarrow x_{F} = \frac{(3 - 2 \sqrt{6}) s}{10}$ $\Rightarrow x_{K} = \frac{(3 + 2 \sqrt{6}) s}{10}$ $Δ x_{G F} = x_{G} - x_{F} = \frac{(2 \sqrt{6} - 3) s}{10}$ $Δ x_{H K} = x_{K} - x_{H} = \frac{(3 + 2 \sqrt{6}) s}{10} - \frac{s}{2} = \frac{(2 \sqrt{6} - 2) s}{10}$ $\frac{w - u}{v} = \frac{Δ x_{H K} - Δ x_{G F}}{Δ x_{E G}}$ $= \frac{\frac{(2 \sqrt{6} - 2) s}{10} - \frac{(2 \sqrt{6} - 3) s}{10}}{\frac{s}{2}} = \frac{1}{5}$ $\Rightarrow v = 5 (w - u) \equiv y = 5 (z - x) i n o r i g i n a l q u e s t i o n ✓$ $e q n . o f J C :$ $y = s + \frac{x - s}{\sqrt{3}} = \frac{x}{\sqrt{3}} + (1 - \frac{1}{\sqrt{3}}) s$ $e q n . o f A E :$ $y = \sqrt{3} x$ $y_{M} = \sqrt{3} x_{M} = \frac{x_{M}}{\sqrt{3}} + (1 - \frac{1}{\sqrt{3}}) s$ $\Rightarrow x_{M} = \frac{(\sqrt{3} - 1) s}{2}$ $e q n . o f B E :$ $y = - \sqrt{3} (x - s)$ $y_{N} = - \sqrt{3} (x_{N} - s) = \frac{x_{N}}{\sqrt{3}} + (1 - \frac{1}{\sqrt{3}}) s$ $\Rightarrow x_{N} = \frac{(4 - \sqrt{3}) s}{4}$ $Δ x_{M J} = x_{M} - x_{J} = \frac{(\sqrt{3} - 1) s}{2}$ $Δ x_{C N} = x_{C} - x_{N} = s - \frac{(4 - \sqrt{3}) s}{4} = \frac{\sqrt{3} s}{4}$ $Δ x_{N M} = x_{N} - x_{M} = \frac{(4 - \sqrt{3}) s}{4} - \frac{(\sqrt{3} - 1) s}{2} = \frac{3 (2 - \sqrt{3}) s}{4}$ $\frac{c - a}{b} = \frac{Δ x_{C N} - Δ x_{M J}}{Δ x_{M N}}$ $= \frac{\frac{\sqrt{3} s}{4} - \frac{(\sqrt{3} - 1) s}{2}}{\frac{3 (2 - \sqrt{3}) s}{4}} = \frac{1}{3}$ $\Rightarrow b = 3 (c - a) ✓$
	Commented by Spillover last updated on 13/Apr/25

	$g r e a t . t h a n k y o u$
	Answered by Spillover last updated on 17/Apr/25

	Answered by Spillover last updated on 17/Apr/25

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