Morley's Theorem
G. Zsolt Kiss' proof

This is the latest proof that was sent (September 02, 2006) to me by G. Kiss Zsolt from Hungary. Like Conway's and Newman's, this one, too, employs the cheating (backward) strategy. Also, one can't miss its relation to Conway's proof. It uses the same add-ons but in a more natural and symmetric manner.

Assume angles a, b, c are given such that a + b + c = 60°. Pick up an equilateral triangle and, on its sides form isosceles trapezoids with base angles 2a, 2b, 2c, respectively, and the sides equal to the top base.

In such a trapezoid B0B1C2C0, the diagonals serve as the bisectors of the base angles. Indeed, assuming ∠B0B1C2 = 2a, ∠B1B0C0 = 180° - 2a. And since, by construction, ΔB1B0C0 is isosceles, its base angles equal

[180° - (180° - 2a)]/2 = a.

Now we form ΔABC by passing three straight lines through the dangling vertices of the trapezoids:

We get three small isosceles triangles, ΔB0B1B2 is one of them. Its apex angle B1B0B2 is found from

∠B1B0B2 = 360° - 60° - (180° - 2a) - (180° - 2c)
 = 2a + 2c - 60°
 = 2(60° - b) - 60°
 = 60° - 2b.

Hence its base angles equal

[180° - (60° - 2b)]/2 = 60° + b,

while the supplementary angle B0B1A = 120° - b. Similarly,

∠C0C2A = 120° - c.

It follows that ∠C0B1A = 120° - a - b so that the two sum up to 180°:

∠C0C2A + ∠C0B1A = 180°.

This means that quadrilateral AC2C0B1 is cyclic. In the same manner, quadrilateral AC0B0B1 is also cyclic. Hence so is the pentagon AC2C0B0B1. Since, in the circumcircle, the three chords B0B1, B0C0, and C0C2 are equal, lines AB0 and AC0 trisect angle at A, which, as easily seen is 3a.

Now trying to reverse the steps, assume in ΔA'B'C' angles equal 3a, 3b, 3c so that a + b + c = 60°. Starting with any equilateral triangle and following the above construction, we obtain ΔABC similar to ΔA'B'C'. Comparing the two we can modify the original equilateral triangle so as to obtain ΔABC equal to the given ΔA'B'C'. Which shows that the equilateral triangle is indeed obtained by Morley's trisectors.

Compared to Conway's or Newman's, it may be just a little longer but does not come even close to using trigonometry.

Hubert Shutrick sent me the following comment (with the entire proof on a separate page):

The third diagram in the proof shows how the vertices of ΔABC can be constructed using the points A1, A2, B1, B2, C1, C2 although A2 and C1 are not shown. Here is another motivation for these six points. Note that it is a straight edge and compass construction. Given the equilateral triangle A0B0C0, the required vertex A has to be such that the angle B0AC0 is a and the locus of points making this angle is the arc of a circle through B0 and C0. Any other chord of the same length will also subtend the angle a for points on the circle. That is why B1 is an interesting point. The angle B0B1C0 is a so B1 is on the circle and, since the length B0B1 is 1, the angle B0AB1 will be a for points A on the circular arc. Hence, if the theorem is true, B1 must be on the side AC of the required triangle.

By symmetry, the six points mentioned above must be on the sides of the triangle ABC. The proof then consists of showing that it's angles are indeed 3a, 3b and 3c, simple angle calculations. There are two special cases that should be mentioned. There is the case a = b = c = 20°, when the theorem is obvious. The other case is when two of a, b, c add up to 40° without being equal. In this case, two of the six points coincide but the other four are distinct so the triangle ABC can still be constructed from them.

I found this proof long ago, in the 1970s, while teaching Euclidean geometry to prospective mathematics' teachers at Karlstad's university college.

Amic has observed that a correction is in order:

I read

  "There are two special cases that should be mentioned. There is the case a = b = c = 20°, when the theorem is obvious. The other case is when two of a, b, c add up to 40° without being equal. In this case, two of the six points coincide but the other four are distinct so the triangle ABC can still be constructed from them."

There is a mistake : the other case is when a or b or c = 30°, and in fact we cannot define the line joining two points which are the same...

And I don't see why the case a = b = c = 20° is a special case...

So in my opinion this proof is wrong for a triangle ABC with a right angle.

My comment is that the proof can be saved by drawing the necessary line not "through two points" but at a certain angle at the "double point". For example, if b = 30° then B1 and B2 coincide and AC should be drawn as to form angles 90° - 2a and 90° - 2c with B1C2 and, respectively, B2A1.

Hubert Shutrick has later added the following:

If B' is the midpoint of B1B2, then B0B' is orthogonal to AC and ∠C0B0B' is 150° + c - a since it is the mean of 180° - 2a and 360° - 60° - (180° - 2c). Similarly, ∠B0C0C' is 150° + b - a and the third exterior angle of the triangle formed by B0B', C0C' and B0C0 is indeed 3a. As for the special case that I bungled, if B1 = B2 = B', the calculation ensures that AC is the line orthogonal to B0B'.

Morley's Miracle

On Morley and his theorem

  1. Doodling and Miracles
  2. Morley's Pursuit of Incidence
  3. Lines, Circles and Beyond
  4. On Motivation and Understanding
  5. Of Looking and Seeing

Backward proofs

  1. J.Conway's proof
  2. D. J. Newman's proof
  3. B. Bollobás' proof
  4. G. Zsolt Kiss' proof
  5. Backward Proof by B. Stonebridge
  6. Morley's Equilaterals, Spiridon A. Kuruklis' proof
  7. J. Arioni's Proof of Morley's Theorem

Trigonometric proofs

  1. Bankoff's proof
  2. B. Bollobás' trigonometric proof
  3. Proof by R. J. Webster
  4. A Vector-based Proof of Morley's Trisector Theorem
  5. L. Giugiuc's Proof of Morley's Theorem
  6. Dijkstra's Proof of Morley's Theorem

Synthetic proofs

  1. Another proof
  2. Nikos Dergiades' proof
  3. M. T. Naraniengar's proof
  4. An Unexpected Variant
  5. Proof by B. Stonebridge and B. Millar
  6. Proof by B. Stonebridge
  7. Proof by Roger Smyth
  8. Proof by H. D. Grossman
  9. Proof by H. Shutrick
  10. Original Taylor and Marr's Proof of Morley's Theorem
  11. Taylor and Marr's Proof - R. A. Johnson's Version
  12. Morley's Theorem: Second Proof by Roger Smyth
  13. Proof by A. Robson

Algebraic proofs

  1. Morley's Redux and More, Alain Connes' proof

Invalid proofs

  1. Bankoff's conundrum
  2. Proof by Nolan L Aljaddou
  3. Morley's Theorem: A Proof That Needs Fixing

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