Euclid's missing theorem

The easiest way to be original is to be complicated. One way to come up with a new theorem in plane geometry is to do something so complicated that ancient geometers would not have understood or cared about your theorem. That kind of theorem isn’t impressive.

What is impressive is a theorem discovered two millennia after Euclid that seems like it would make an ancient geometer ask “Why didn’t I think of that?”

The Erdős-Mordell triangle theorem is such a theorem.

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The natural logarithm is negative on the interval (0, 1) and goes off to negative infinity on the left end. Take the graph of this function, and move every point up by the smallest even number that makes the value positive.

It would seem that this function would be hard to integrate, either analytically or numerically. But the analytical integral has a simple expression. And numerical integration software does better than I would have expected without giving the integrator some help.

More here in the post on the logarithmic sawtooth.

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Thanks for reading.