Monday, March 10, 2008

Mathematics for Arithmeticians: Pons asinorum

In today's post, I will show Pappus's proof of the so-called Pons asinorum or Bridge of Asses. The theorem is given this name because it is the first difficult theorem in Euclid's Elements. The theorem states that the base angles of an isosceles triangle are equal.

Consider triangle ABC, with AB = BC. Now we want to prove that angle A is equal to angle C. One useful trick in geometry for proving things equal is to use congruence of triangles, but we only have one triangle.

Or do we? Pappus' trick is to look at one triangle two ways: ABC and its reflection CBA. AB = CB (by our starting assumption), BA = BC (by the same assumption), and AC = CA (they're the same line). So ABC ≡ CBA (all three corresponding sides are equal, so they're congruent). Since they're congruent, the corresponding angles are equal: angle A equals angle B.

That's it. This trick allows us to prove the theorem in just a few steps.

Why do I think this elegant? By looking at one thing in two ways, we learnt something new, and remarkably quickly. And the great thing is that you can use this trick again, to prove the converse using angle-side-angle congruence. And a trick you can use more than once is a technique.

If you followed this, well done. You have crossed the Bridge of Asses.

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