Study Limitations The first four proofs presented in the paper have a notable limitation: they don’t work for isosceles right triangles (triangles where both non-right angles are 45 degrees).

From such assumptions Euclid went on to show, for example, that there are infinitely many prime numbers, and that the angles at the base of an isosceles triangle are equal.

It then details the original proof presented at the conference, which applies to all non-isosceles right-angled triangles, and adds four others also made using trigonometry.

I checked it out, and the proofs are correct. Here's the Isosceles Lemma: take an isosceles triangle (two of the sides have the same length) with no angle larger than 90 degrees.

For the next few weeks, the planet Mars will form an almost perfect isosceles triangle with the two bright stars marking the heads of the Gemini Twins, Pollux and Castor.

Learn about and revise the different angle properties of circles described by different circle theorems with GCSE Bitesize AQA Maths.