Thales and Pythagoras are quasi-mythical figures, and we don't actually know anything concrete about any mathematical accomplishments they might have had, which

Thales and Pythagoras are quasi-mythical figures, and we don't actually know anything concrete about any mathematical accomplishments they might have had, which are all apocryphal and date from many centuries after their deaths. Greek deductive mathematics per se dates from at least a century after Thales' time, while many of the basic facts about Euclidean geometry were understood in ancient Egypt and Mesopotamia long before him, and it is most likely that Thales himself never did any of the mathematical or scientific things attributed to him. Viktor Blåsjö's speculation that ancient Greeks began with the same insight as Lockhart's 7th grade student is plausible but is not backed by any evidence whatsoever. (This is an insight that many people have had over the centuries, certainly including anyone deeply investigating cyclic quadrilaterals, but also probably plenty masons or metalworkers working with circles and right angles, etc.) > sum and difference of radius vectors This is a nice one. Another way to use Thales' theorem in characterizing a circle, without involving the center point, is to start with one point P on a circle and a vector d which is a diameter from that point to the antipodal point. Then the vector v from P to any other point Q on the circle satisfies v² = v · d, or equivalently v · (v − d) = 0.