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Three lines are given. Find three points on these lines, one point on each line, that are vertices of an equilateral triangle

Well if I consider a point on one line my goal would be to construct the other two points. If the lines are intersecting all at one point perpendicular to each other. Could I then use a circle to construct the other points? I'm a little unsure how to visualize this problem. Any hints would be appreciated.

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    $\begingroup$ Cosider cases. One case is all lines intersect at a point. Second case all lines are parallel. Last case the lines form the sides of a triangle and an equilateral triangle can be inscribed in it. $\endgroup$
    – Somos
    Commented Nov 28, 2017 at 21:28
  • $\begingroup$ ... (continuation of @Somos) ...as is established in (math.stackexchange.com/questions/186432/…) $\endgroup$
    – Jean Marie
    Commented Nov 28, 2017 at 22:15
  • $\begingroup$ I forgot another case with two parallel lines cut by a transversal line. $\endgroup$
    – Somos
    Commented Nov 29, 2017 at 2:08

1 Answer 1

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Let the three lines be $r,s,t$. Let $A\in r$.

  1. Rotate $s$ by $60^\circ$ counter-clockwise around $A$ to get $s'$;
  2. Define $B$ as $s'\cap t$;
  3. Define $C$ as the intersection between the perpendicular bisector of $AB$ and $s$.

$ABC$ is equilateral. Can you see why?

enter image description here

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