Thales' theorem

The story goes that Thales measured the height of the Great Pyramid of Egypt with... its shadow! His stroke of genius: under the sun, a stick and the pyramid form two triangles of the same shape, one small, one giant. Same shape means proportional sides: that is the whole spirit of Thales' theorem.

Here is the classic setup. In a triangle ABC, place a point M on side [AB] and a point N on side [AC]. If line (MN) is parallel to line (BC), then the small triangle AMN is a scaled-down copy of the big triangle ABC. All lengths are multiplied by the same factor, and we get the equality of three ratios: AM/AB = AN/AC = MN/BC.

How do you use it? Write the ratios, spot the complete one (both lengths known) and the one holding the unknown, then solve with cross multiplication. A life-saving tip: always put the SMALL triangle's lengths on top and the BIG triangle's lengths at the bottom. Mixing the floors is the most common mistake!

Mind the entry condition: no parallel lines, no Thales! It is the parallelism of (MN) and (BC) that guarantees the two triangles have the same shape. The theorem also exists in the butterfly configuration, when the lines cross at a point between the two parallels: the ratios are then written from the crossing point.