Equilateral Triangle on Three Lines

What is this about?

Problem

Given three straight lines (denoted below by two points $AB,$ $CD,$ $EF$).

Equilateral Triangle on Three Lines - problem

Construct an equilateral triangle with vertices one per line.

Construction

Choose an arbitrary point on one of the line, say $X$ on $EF.$ Rotate $CD$ around $X$ $60^{\circ}$ into $C'D'.$ Let $Y'$ be the intersection of $C'D'$ with $AB$ and $Y$ the point that was mapped into $Y'$ by the rotation.

Equilateral Triangle on Three Lines - solution

Then $\Delta XYY'$ is equilateral and $X\in EF,$ $Y'\in AB,$ and $Y\in CD.$

In general, the solution is not unique, e.g.,

Equilateral Triangle on Three Lines - extra example

Acknowledgment

The construction is classic. For three parallel lines (when the solution is unique up to isomorphism), it can be found in I. M. Yaglom, Geometric Transformations I (MAA, 1962), problem 18. The latter problem has been discussed elsewhere.

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