### Shape and Territory

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

### 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.)

# Impossible Triangles?

### Why do this problem

This problem will make students experiment, conjecture and prove. They will need to understand the interplay between rational and irrational numbers.

### Possible Approach

This activity lends itself to hands-on experimentation with cut-outs of triangles before algebraic proof is attempted. Without cut-out shapes it is a good exercise in visualistion. Students might need to be encouraged to pursue various lines of enquiry.

Explaining a proof clearly to someone else is a very good way to discover where the holes in the argument are. You could ask students to explain their proofs to each other in pairs and then to the whole class. Can the class point out areas where proofs or explanations are unclear?

### Key Questions

How do we relate side-length and area of triangles?
Which variables in the problem are rational and which are irrational?
Can we make use of the angle properties of the triangles?

### Possible Extensions

Can you suggest other regular shapes which you could prove to be impossible?

### Possible Support

Try the easier question Equal Equilateral Triangles first.
Struggling students could be asked to hear other students proofs and comment on their clarity.