### Areas and Ratios

Do you have enough information to work out the area of the shaded quadrilateral?

### 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?

### The Root Cause

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

# Shape and Territory

##### Age 16 to 18 Challenge Level:

Why do this problem?
It provides practice in using the tan formula and an opportunity to reflect on the bigger picture beyond school mathematics.

Possible approach
The result can be proved without using the fact that the angles add up to 180 degrees and this could be taken up as a challenge.

Then ask "What is the 'territory' this result belongs to?"

Sue's proof given here uses the fact that the angles of the triangle $ABC$ add up to 180 degrees. However it is just as easy to prove this result without using the fact about the sum of the angles of the triangle so it must be true for triangles 'living in other geometries' where the angles of triangles do not add up to 180 degrees such as Spherical Geometry.

The lines in Spherical Geometry are great circles on the surface of the sphere. By drawing lines like the lines of longitude and the equator on the earth you will soon be able to convince yourself that spherical triangles have angle sums greater than 180 degrees.

This expression gives a certain property for triangles for all 3 geometries, for Euclidean Geometry where the angles of triangles add up to 180 degrees, for Spherical (also called Elliptical) Geometry where the angles of triangles add up to more than 180 degrees and for Hyperbolic Geometry where the angles of triangles add up to less than 180 degrees.

Key question
Here we have $(A-B)$, $(B-C)$ and $(C-A)$. How can we write this using only 2 variables?

Possible extension
See the articles Strange Geometries which is accompanied by a lot of ideas for project work suitable for 12 year olds and older students on Non-Euclidean Geometries.