If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0
what can you say about the triangle?
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
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
Proof is, of course, a central part of mathematics. However constructing proofs is often difficult for novices. This problem provides a bridge through the device of classic faulty proofs of 'obviously' wrong results. Analysing these faulty proofs will provide training in reading proofs, raise awareness of mathematical hazards, such as division by zero, and provide motivation for rigour. You
might wish to use some of these proof every so often throughout the year.