Proportional representation, proportional taxation, mean proportional, fourth proportional: all these expressions use the same word, "proportional." We may have a more or less precise idea of what it means, but do we know its many applications in geometry—and, in particular, that it underpins the proofs of most of the great theorems?
Geometry books are full of ratios, and equal ratios mean proportionality. In geometry, the first theorem that comes to mind when proportionality is mentioned is Thales’ theorem. It is named in honor of the illustrious Greek mathematician of the 6th century BCE, who is said to have calculated the height of a pyramid with a stick, using proportions. But its earliest known proof appears in Euclid’s The Elements, written around three centuries BCE (Proposition 2, Book VI; see "With Euclid").
The gentle flow of similarity
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But in geometry, "proportions" also means "similar triangles"—more precisely, triangles whose corresponding sides are proportional. This idea recurs in many proofs, even beyond the setting of triangles, as in Ptolemy’s theorem.
The theorem gives a necessary and sufficient condition for four points A, B, C and D to be concyclic: the equality AC × BD = AB × CD + BC × AD must hold.
