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# Pareq Exists

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### Doodles

### Russian Cubes

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Age 14 to 16

Challenge Level

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?