Morley's Trisector Theorem/Dijkstra's Proof

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Theorem

Let $\triangle ABC$ be a triangle.

Let the internal angles of $\triangle ABC$ be trisected.

Let the points where these angle trisectors first intersect be $D$, $E$ and $F$.

Morleys-Theorem.png


Then $\triangle EDF$ is equilateral.


Dijkstra's Proof

Choose angles $\alpha$, $\beta$ and $\gamma$ all greater than $0$ such that $\alpha + \beta + \gamma = 60 \degrees$.

Draw an equilateral triangle $\triangle XYZ$.


Construct $\triangle AXY$ and $\triangle BXZ$ with the angles as indicated:

Morleys-Theorem-Dijkstra-Proof.png


We have that:

\(\ds \angle AXY\) \(=\) \(\ds 60 \degrees + \beta\)
\(\ds \angle AYX\) \(=\) \(\ds 60 \degrees + \gamma\)
\(\ds \angle BXZ\) \(=\) \(\ds 60 \degrees + \alpha\)
\(\ds \angle BZX\) \(=\) \(\ds 60 \degrees + \gamma\)


Because $\angle AXB = 180 \degrees - \paren {\alpha + \beta}$ , it follows that:

if $\angle BAX = \alpha + x$ then $\angle ABX = \beta - x$

Using the Sine Rule $3$ times, in $\angle AXB$, $\angle AXY$ and $\angle BXZ$, we have:

\(\ds \dfrac {\map \sin {\alpha + x} } {\map \sin {\beta - x} }\) \(=\) \(\ds \dfrac {BX} {AX}\)
\(\ds \) \(=\) \(\ds \dfrac {XZ \map \sin {60 \degrees + \gamma} / \sin \beta} {XY \map \sin {60 \degrees + \gamma} / \sin \alpha}\)
\(\ds \) \(=\) \(\ds \dfrac {\sin \alpha} {\sin \beta}\)

In the range in which these angles lie, the left hand side of the above is a strictly increasing function of $x$.

Thus we conclude that $x = 0$.

The result follows.

$\blacksquare$


Source of Name

This entry was named for Edsger Wybe Dijkstra.


Historical Note

This proof was published by Edsger Wybe Dijkstra on $30$ December $1975$ in an open letter to Ross Honsberger.

As he put it:

the other day I encountered your delightful booklet "Mathematical Gems". On account of Chapter 8, I concluded that you might be interested in the following proof of Morley’s Theorem "The adjacent pairs of the trisectors of a triangle always meet at the vertices of an equilateral triangle." ... I found this proof in the early sixties, but am afraid that I did not publish it.


Sources