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Question Number 166701 by ajfour last updated on 25/Feb/22

	Answered by mr W last updated on 26/Feb/22

	$B C = 1$ $C D = x$ $C E = \frac{x}{2}$ $D E = \frac{\sqrt{3} x}{2}$ $A E = 1 - \frac{x}{2}$ $A D = \sqrt{1^{2} + x^{2} - 2 \times 1 \times x \times \cos \frac{π}{3}} = \sqrt{x^{2} - x + 1}$ $\frac{F E}{A E} = \frac{D E}{A D}$ $F E = \frac{\sqrt{3} x (1 - \frac{x}{2})}{2 \sqrt{x^{2} - x + 1}}$ $\frac{A F}{A E} = \frac{A E}{A D}$ $A F = \frac{{(1 - \frac{x}{2})}^{2}}{\sqrt{x^{2} - x + 1}}$ $B E = \sqrt{1^{2} + {(\frac{x}{2})}^{2} - 2 \times 1 \times \frac{x}{2} \times \cos \frac{π}{3}} = \sqrt{\frac{x^{2}}{4} - \frac{x}{2} + 1}$ $B F = \sqrt{\frac{x^{2}}{4} - \frac{x}{2} + 1} - \frac{\sqrt{3} x (1 - \frac{x}{2})}{2 \sqrt{x^{2} - x + 1}}$ ${B F}^{2} + {A F}^{2} = 1^{2}$ ${(\sqrt{\frac{x^{2}}{4} - \frac{x}{2} + 1} - \frac{\sqrt{3} x (1 - \frac{x}{2})}{2 \sqrt{x^{2} - x + 1}})}^{2} + {(\frac{{(1 - \frac{x}{2})}^{2}}{\sqrt{x^{2} - x + 1}})}^{2} = 1$ $x \approx 0.3542$ $\frac{B D}{C D} = \frac{1 - x}{x} \approx 1.8233$
	Commented by Tawa11 last updated on 26/Feb/22

	$Great sir$
	Commented by ajfour last updated on 26/Feb/22
	Thanks sir. I got it exact easily somehow!
	Commented by mr W last updated on 26/Feb/22

	$v e r y n i c e s o l u t i o n!$
	Answered by ajfour last updated on 26/Feb/22

	Commented by ajfour last updated on 26/Feb/22

	$l e t B C = A B = A C = 1$ $B D = p, C D = q$ $\cos (60 ° - θ) = p \cos θ$ $\frac{q \sqrt{3}}{2} \sin ϕ = \frac{q \sqrt{3}}{2} \sin (30 ° - θ) = p \sin θ$ $\frac{q}{p} = \frac{2}{\sqrt{3}} (\frac{2}{\cot θ - \sqrt{3}}) = m . . . (i)$ $\Rightarrow \sqrt{3} \cot θ = \frac{4}{m} + 3 . . . (i)$ $p + q = 1 \Rightarrow 1 + m = \frac{1}{p}$ $\Rightarrow 1 + m = \frac{2}{1 + \sqrt{3} \tan θ}$ $\Rightarrow \sqrt{3} \tan θ = \frac{2}{1 + m} - 1 . . . (i i)$ $f r o m (i) \times (i i) w e o b t a i n$ $(\frac{2}{1 + m} - 1) (\frac{4}{m} + 3) = 3$ $\Rightarrow (1 - m) (4 + 3 m) = 3 m (1 + m)$ $\Rightarrow 6 m^{2} + 4 m - 4 = 0$ $\Rightarrow 3 m^{2} + 2 m - 2 = 0$ $o r m = \frac{\sqrt{7} - 1}{3}$ $\Rightarrow \frac{1}{m} = \frac{B D}{C D} = \frac{\sqrt{7} + 1}{2} \approx 1.82287566$
	Commented by Tawa11 last updated on 03/Mar/22

	$Great sir$
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