### 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?

### 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.

### 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:
• Draw any triangle P, Q, R.
• Find points A, B and C, one on each side of the triangle (but not at the vertices), such that the area of triangle ABC is one quarter the area of triangle PQR.

• Now find points such that the area of triangle ABC is one half the area of triangle PQR and a method that will give any fraction of the area between one quarter and one.
• How would you adapt your method to give fractions less than one quarter?