[Update: Dan Lawson has proved the theorem without trigonometry. Thanks, Dan!]
I spent a good chunk of last week reading about David Johnson Leisk (1906–1975), who is better known by his nom-de-plum Crockett Johnson. Johnson is most well known as the author of Harold and the Purple Crayon, a children’s book from 1955, and its sequels. Johnson was also the author of the 1940s comic Barnaby.
Later in his life Johnson became interested in mathematics. He was particularly interested in geometry, and most specifically in the problems of antiquity (squaring the circle, trisecting the angle, doubling the cube, and constructing regular polygons). He turned many geometric theorems into works of art. Eighty of his paintings are now at the Smithsonian.
Johnson even created new mathematics. I would like to discuss one of his contributions here. (See his article “A Construction for a Regular Heptagon,” The Mathematical Gazette Vol. 59, No. 407 (Mar., 1975), pp. 17-21.)
The heptagon is notewothy because it is the regular polygon with the fewest number of sides that cannot be constructed with compass and straightedge alone. In his article, Johnson gives a way to construct the heptagon using a marked straightedge (this is called a neusis construction). Johnson did not give the first neusis construction of a heptagon—François Viète gave the first such construction in 1593. (Also, Archimedes gave an unorthodox neusis-like construction).
However, Johnson’s proof used trigonometry (including the law of cosines and several trigonometric identities). My question to you is: Is there a purely geometric proof of his result? I played around with it for a little while and couldn’t find one, and I couldn’t find a geometric proof in the literature.
The key to Johnson’s construction is producing 3:3:1 triangle; that is, a triangle in which the angles are in a 3:3:1 ratio (they would be and The three vertices of the triangle are three vertices of a regular heptagon. If we construct the circumcircle, then it is easy to construct the four remaining vertices with a compass and straightedge.
Here’s his construction of a 3:3:1 triangle using a marked straightedge—that is, an otherwise ordinary straightedge, but possessing a mark one unit from the end (or equivalently, two marks one unit apart). We begin by drawing a line segment AB of length one (see below, left). Construct a unit line segment AC perpendicular to AB. Also, construct the perpendicular bisector to AB; call it l. Then construct a circle with center B and radius BC. Now we perform the neusis construction with the marked straightedge: Construct a line AD so that D is on l and D is one unit from the circle. (That is, the end of the straightedge is at D, the mark is on the circle, and the edge passes through A.) Then is the 3:3:1 triangle.
A geometric proof?
Johnson’s proof that is a 3:3:1 triangle used trigonometry (see his article for details). Is there a geometric proof?
Boiled down to its essence, here’s the question: Suppose is isosceles and E is on AD. Moreover, suppose and , prove that
Here is a fact that may help. Johnson discovered this fact—ironically—when he was sitting in a café in Syracuse on Sicily (the birth place of Archimedes) playing with the wine and food menus and some tooth picks. If we have an isosceles triangle ABD with the point E on BD and F on AD such that then triangle ABD is a 3:3:1 triangle. (This is easy to prove. Suppose Then, because is isosceles, . So the exterior angle of is Because is isosceles, Lastly, observe that and are similar, so It follows that )
Thus, a possible route to proving the theorem is to find a point F in the original diagram so that
If you find a proof, I’d love to hear about it!
Here are two of the paintings Johnson made from his work with the heptagon.