Morley's Theorem (Haverford 1899)
Morley's Theorem is a Haverford classic, having been discovered in 1899 by Frank Morley, who was a professor at Haverford (and also the father of writer Christopher Morley). It states:
The three points of intersection of the adjacent angle trisectors of any triangle form an equilateral triangle.
For more information about Morley's Theorem, click on the following:
Several years ago I wrote a Mathematica program to generate an animated gif file illustrating Morley's Theorem. The program had a slight bug which produced the interesting result you see above. When the moving vertex dips below the line formed by the other two vertices, one set of angle trisectors becomes the trisectors of the external angle at that vertex. Amazingly, an analog of Morley's Theorem continues to hold: the three pairwise intersection points form an equilateral triangle.
This hybrid analog of Morley's Theorem suggested that there might be other versions as well, and indeed there are. For example:
The three points of intersection of the adjacent external angle trisectors of any triangle form an equilateral triangle.
Note that in forming the intersections, the external trisectors must all be followed in a retrograde direction. Here is a picture showing both versions of the theorem together:
And finally, a picture showing the internal, external, and hybrid versions all in one picture:
Stay tuned: I plan to make a movie based on the last picture!
Epilog:
After making these observations, which resulted from a serendipitous computer blunder, I began to dimly recall several conversations which I had in 1978 upon arriving at Haverford with one of my predecessors, Cletus O. Oakley, who had retired from Haverford some years before. He was writing a survey article on Morley's theorem for the American Mathematical Monthly:
At the time I did not fully grasp the diagrams in that paper, but I recalled them as vaguely similar to the pictures I was drawing with Mathematica. With 20-20 hindsight, I now see that they show all of the variations in Morley's theorem that I had recently discovered, and more. In fact the massive historical bibliography in Oakley's paper shows that these facts were rediscovered by many authors, beginning as early as 1910.
Oakley notes that there are actually 18 different equilateral triangles formed by various intersections of trisectors, and that in all cases the corresponding sides are parallel.
It is difficult to discover truly new results in mathematics. It is all the more difficult (and extraordinarily rare) to find new theorems in elementary geometry, e.g. about equilateral triangles. But that's what Frank Morley did in 1899. What must have gone through his mind when he first noticed that something like Morley's theorem might be true?
Updated 8/20/02