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Napoleon's Hat

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

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Plane to See

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.

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Five Circuits, Seven Spins

A circular plate rolls inside a rectangular tray making five circuits and rotating about its centre seven times. Find the dimensions of the tray.

Halving the Triangle

Stage: 5 Challenge Level: Challenge Level:1

Imagine cutting off three triangles and finding the area of the remaining triangle, that is triangle ABC. If each side of triangle PQR is divided in the same ratio, say t:1-t, then can you find the areas of all 4 triangles in terms of t?