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

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.

• 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. Commented Nov 28, 2017 at 21:28
• ... (continuation of @Somos) ...as is established in (math.stackexchange.com/questions/186432/…) Commented Nov 28, 2017 at 22:15
• I forgot another case with two parallel lines cut by a transversal line. Commented Nov 29, 2017 at 2:08

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?