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Question Number 165150 by ajfour last updated on 26/Jan/22

Commented by ajfour last updated on 26/Jan/22

$T o f i n d e s p e c i a l l y O B, m a k e t h i s$ $c o n s t r u c t i o n p o s s i b l e .$
	Answered by aleks041103 last updated on 27/Jan/22

	$N = l \cap B Q, l ⊥ A B, O \in l$ $l e t O B = a :$ $△ B O N \sim △ B A Q$ $\Rightarrow \frac{A Q}{O N} = A Q = \frac{c + a}{a} = \frac{A B}{O B}, w h e r e O N = 1$ $A Q = B S \Rightarrow B S = 1 + \frac{c}{a}$ $∡ B P S = 90 ° \Rightarrow \frac{B S}{B O} \frac{O N}{O B} = 1$ $\frac{c + a}{a^{2}} = \frac{a}{1} \Rightarrow a^{3} - a - c = 0$ $\Rightarrow F o r t h i s c o n s t r u c t i o n t o b e p o s s i b l e :$ ${O B}^{3} - O B = c$ $o r$ $O B = \sqrt[3]{\frac{2}{27 c + 3 \sqrt{3} \sqrt{27 c^{2} - 4}}} + \sqrt[3]{\frac{9 c + \sqrt{3} \sqrt{27 c^{2} - 4}}{18}}$
	Commented by ajfour last updated on 27/Jan/22

	$t h a n k s s i r, h o p e t o g e t s o m e$ $c o n s t r u c t i o n w a y t o o . .$
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