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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 
The shortest path between any two points on a snooker table is the straight line between them but what if the ball must bounce off one wall, or 2 walls, or 3 walls?
P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.