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This is a beautiful result involving a parabola and parallels.

Napoleon's Hat

Age 16 to 18 Challenge Level:

The diagram shows three equilateral triangles $ABC, AYX$ and $XZB$. The point $X$ is a moveable point on $AB$. The points $P$, $Q$ and $R$ are the centres of the three triangles.

Experiment with the dynamic diagram.

What can you say about triangle $PQR$?

Can you prove your conjecture?

Created with GeoGebra

You can download your own free copy of Geogebra from the above link and draw your own dynamic diagrams.