
Re: A Theorem concerning the Trisectors of a Triangle
Posted:
Sep 15, 1998 4:31 PM


Before the facts are buried in the sands of time, it occurs to me that I should write a Hysterical Note.
On 931015 Joseph Goggins submitted the following problem to the Monthly
Prove that every prime, p > 7, of the form 3n+1 can be written as the sixth root of a^2 + 4762800 b^2 for a unique choice of natural numbers a and b
which was sent on to me to referee. Needless to say, I did not encourage the editors to use it. No solution was given, it was based on experiment, but those I consulted said that it was a suitable exercise for a graduate student in the area.
Fortunately, Goggins did indicate in a covering letter where he got the problem from. He'd been looking for, and finding, triangles with rational sides and a rationalsided (`canonical') Morley triangle. This grabbed me, and in trying to find the general solution, I soon realized that algebraically I was solving cubic equations and geometrically I was drawing sets of three trisectors, so that one might expect 27 Morley triangles. For various reasons, there are in fact only 18, but that was exciting enough. I rang Coxeter, who happened to have Willy Moser visiting him, and neither of them knew this. I told John Conway, who said that funnily enough, he had just made the same discovery.
So I wrote a joint paper, discovering the Lighthouse theorem the while, and submitted it to the Monthly. One referee responded fairly quickly, saying, `OK but there are several things and references missing' (including, it emerged, the fact that Morley knew that there were 18 Morley triangles). A second referee said `isn't this like a paper I refereed for you a year or so ago?' at which point the Editor, finding this to be rather a strange case, stepped in, since the earlier paper was by the first referee, who hadn't got around to making the recommended amendments. So it is now hanging around as a semiaccepted triple paper, which I hope to (see) complete(d) before I die.
Meanwhile, back at the ranch, I got Andrew Bremner to help me find the parametrization of the rational Morley triangle problem and it emerged that if one Morley triangle was rational, so were the other 17. In fact so were the other 71, where one notes that the triangles ABC, HBC, AHC, ABH all have the same EulerFeuerbach ninepoint (or twelvepoint, since JHC has thrown in three more points) circle, and the 18 Morley triangles from each are all homothetic. Mike Guy gave a better exposition than ours and pointed out that we had missed the case of the equilateral triangle, just 6 of whose 18 Morley triangles are congruent to the original triangle.
So we now have a quadruple paper (Bremner, Goggins, Guy, Guy) on rational Morley triangles. In putting together the final (?) draft, Andrew discovered that we'd missed yet another exception. There's a family of Pythagorean triangles, just two of whose 18 Morley triangles are rational!
Here concludeth, almost, the research announcement. R.

