Pythagoras, Euclid, Archimedes and a new Trigonometry

Pythagoras ’ theorem, Euclid’s formula for the area of a triangle as one half the base times the height, and Heron’s or Archimedes ’ formula are amongst the most important and useful results of ancient Greek geometry. Here we look at all three in a new and improved light, giving a dramatically simpler and more elegant trigonometry. Three great theorems There are three classical theorems about triangles that every student meets. We work with a triangle A1A2A3 with side lengths d1 = |A2, A3|, d2 = |A1, A3 | and d3 = |A1, A2|. Pythagoras ’ theorem The triangle A1A2A3 has a right angle at A3 precisely when d21 + d 2 2 =