Recently, in a moment of idleness, I fired up ChatGPT and asked "Will you show me some interesting geometric theorems?". It produced some theorems I already knew but two were new to me.
The first was Viviani's theorem. Choose any point P within an equilateral triangle and drop perpendiculars to each side of the triangle. Then the sum of the perpendicular distances is independent of the point chosen.
I didn't know whether this was a very deep theorem (on a par with Morley's miracle theorem for example) or not. However a few minutes of elementary calculations showed me it was easy to prove and I announced it to my daughter Laura when I saw her next. Laura is not a mathematician but she is certainly clever and it was disconcerting that she thought for a only moment before saying "Oh yes, I see". Humiliatingly, her proof was even easier than mine. The triangle is the decomposition of 3 triangles, each with a vertex at P and their combined areas are the sum (base*height)/2. The bases are constant and the sum of the heights is therefore a constant.
Anyway, never let a good proof go to waste. By a similar argument we can replace the equilateral triangle by any convex polygon with equal side lengths. There are other more complicated generalisations.
The second theorem that ChatGPT told me about was Miquel's theorem and not quite so transparent. Let points A, B, C define a triangle (any triangle) and choose any points on the sides: D on side BC, E on side CA, F on side AB (they don't even have to be interior to the side). Now draw the circumcircles of the triples AFE, BDF, CED. The conclusion, Miquel's 3 line theorem, is that these circles pass through a common point M.
This seemed extraordinary given the freedom to choose the 6 points. However, when one knows the proof (an angle-chasing argument) it loses its mystery. It turns out that there are a number of connected results but one of them is also called Miquel's theorem. One begins with a complete quadrangle - a quadrilateral on 4 points A, B, C, D - where opposite sides are extended to meet at two further points E, F.
In this figure of 4 points and 6 lines there are 4 triangles AED, ABF, BEC, DCF. Miquel's 4 line theorem is that the circumcircles of these 4 triangles have a common point M. Again, a seemingly extraordinary result. But, again, there is an angle-chasing argument that demystifies it although the argument is more complex because the figure is more cluttered.
I wondered whether these two theorems were related; the most likely possibility was that the 3 line theorem was a special case of the 4 line theorem. However I couldn't make this work and so I turned once more to ChatGPT and asked it for a derivation of the 3 line theorem from the 4 line theorem. Its response was immediate but, as I soon checked, it had errors. I had seen repeated failures before and had previously found that Claude was less dogmatic. However Claude, while initially optimistic, also failed. However Claude was more humble in admitting its shortcomings and, at one point, while giving an argument observing for itself that it was flawed.
A few days later I did find a connection between the 3 line and 4 line versions. To my surprise, rather than deriving the 3 line version from the 4 line version, I found that two applications of the 3 line version gave rise to the 4 line version. Consider the 3 line version in which the points D, E, F are collinear:
Applying the 3 line theorem directly the conclusion is that the circumcircles of triangles AFE, BDF, and CED have a common point P. But these triangles are three out of the four triangles in the quadrangle shown. Similarly, therefore, the circumcircle of the remaining triangle ABC must pass through this common point also.
In retrospect I believe that the "reasoning" behind ChatGPT's and Claude's failures was syntactic rather than semantic; they put together plausible chains of statements which were plausible because they had the right form. Not surprising in view of the fact that Large Language Models are trained on textual samples where only syntax is present. However my interactions with them were not wasted - they kept me engaged and led me to a conclusion I might not otherwise have drawn.
