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.)

Proof of Pick's Theorem

Stage: 5 Challenge Level:

 We are allowed to assume that any polygon, convex or not, can be split into a finite number of non-overlapping triangles. However in this proof we assume that the interior of the polygon does not have any holes like the red polygon shown with a yellow hole in the diagram. Pick's formula is related to Euler's formula and ${\rm area }(P) = i + {1\over 2}p - q$ where $q$ depends on the number of holes.