### 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.

# Areas and Ratios

##### Age 16 to 18 Challenge Level:
In the diagram below (which is not drawn to scale), the area of the three triangular regions is given.

Perhaps it feels as though there isn't enough information to solve this problem!
If you are stuck, click below to reveal some suggestions that may help you.

First suggestion:

In the GeoGebra applet below, you can move the dots to create different triangles with bases which lie on the same straight line and with a shared apex.

What is the relationship between the ratio of the two areas and the ratio of the two lengths?
Can you explain why?
Can you apply this to the original problem?

Second suggestion:

The red line is half the length of the blue line. Numbers and letters inside a triangle represent its area. What can you say about $A+B$?

Third suggestion:

It might be useful to label the points in the diagram. You could also split the quadrilateral into two triangles, and label them:

Can you use the ideas from the first two suggestions to write down some relationships in terms of P and Q?