It will be seen in this talk that certain geometrical theorems may be proved rigorously by checking only three cases. The idea is what Doron Zeilberger calls an 'Ansatz' -- that once we know the general form of a solution we can find the exact solution by checking a few cases. He gives examples where formulae usually established by induction can in fact be proved by checking a small number of cases. We shall do the same for Napoleon's Theorem and also for geometric theorems which don't seem to have been known either to the ancients or to Napoleon Bonaparte.

High-school mathematics only will be assumed. Themes such as computer algebra and notions of proof will be touched on, as will the historical context of ideas such as calculus and complex numbers seen in first-year university mathematics courses.