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3 parallel lines + triangle

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines. (In the problem 'Pareq Calc' the existence of the equilateral triangle was assumed.)

If you have Java enabled, it may help to use the dynamic diagram below which shows three parallel lines and the fixed point A on one of the lines. The points B and C are free to move along the other two parallel lines in such a way that the lengths AB and AC are always equal.

Click and drag the red points.