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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.
Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?
I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?
Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?