Thales of Miletus

In one of the recent posts we showed you how to get a right angle out of a circle, thanks to this guy. A curious reader mentioned it would be interesting to see the proof. Well you can prove this theorem using trig or algebra–the operative word here being you. I’ve long since forgotten how to work these in any reasonable amount of time. Instead, here’s the way an artisan could do it using a simple geometric construction:


After creating the right triangle above the diameter line AB, make a mirror of this triangle below the line by transferring the leg BC (via your dividers) to establish point D.


Complete the triangle ADB.


Now draw in a second diameter line (CD). Because the two diameters are obviously the same length, and because the other legs are equal to each other (short = short; long = long) this construction is a rectangle: a polygon with four sides meeting at right angle. No matter where C and D occur opposite each other on the rim of the circle, they will always produce this figure. That’s proof enough for this carpenter!