Morley's Theorem for Triangles

Q: What does this picture have to do with the Haverford mathematics department?

A: It illustrates Morley's Theorem. F. Morley was a member of Haverford's mathematics department in the early part of the twentieth century. He is credited with being the first to arrive at the following result (around 1899):

The three points of intersection of the adjacent trisectors of any triangle form an equilateral triangle.

You can find animated representations of Morley's theorem at:

http://www.cut-the-knot.com/triangle/Morley/

http://www.cs.princeton.edu/~ah/alg_anim/version2/Morley.html

and http://members.aol.com/Windmill96/morley/morley.html

Prof. Greene also wrote a Mathematica Notebook which produced an animated example of Morley's Theorem...however due to a small error in the code it produced a surprising (and somewhat serendipitous) result. As one of the vertices rotates to the other side of the line segment generated by the other two vertices, a new triangle is formed (not the intersection of the trisectors), and this triangle still is (at least to the naked eye) equilateral! Check it out.

Morley circulated the result among friends, and it soon spread over the world as mathematical gossip. The first publication of the theorem and proofs came about ten years later, although Morley himself didn't publish "his" result until 1929.

 

The theorem is strikingly beautiful. What is particularly interesting is that it was overlooked for so many hundreds of years since geometry was formalized by Euclid. One possible explanation lies in the fact that angle trisectors cannot be constructed using the simple rules of Euclidean geometry (a straightedge and compass) so perhaps general results concerning trisectors were deemed uninteresting.

Morley's son, Christopher, is perhaps better known for his novels (Thunder on the Left) and his poetry:

A circle is a happy thing to be--
Think how the joyful perpendicular
Erected at the kiss of tangency
Must meet my central point, my avatar.
And lovely as I am, yet only 3
Points are needed to determine me.

 

 

For further reading about Morley's Theorem (and its proof) consult:

Introduction to Geometry, H.S.M. Coxeter, John Wiley & Sons (1969)

Mathematical Questions and their Solutions from the Educational Times (New Series), 15 (1909), pp. 23-24, 47.

American Journal of Mathematics, F. Morley, 51 (1929), pp. 465-472.

Mathematical Reflections, various authors, Cambridge University Press (1970), pp. 177-188.


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This page maintained by jtecosky@haverford.edu. Last updated 7/16/96