Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Topic: A Theorem concerning the Trisectors of a Triangle
Replies: 24   Last Post: Nov 10, 1998 12:19 AM

 Messages: [ Previous | Next ]
 John Conway Posts: 2,238 Registered: 12/3/04
Re: A Theorem concerning the Trisectors of a Triangle
Posted: Sep 15, 1998 1:03 PM

On Tue, 15 Sep 1998, Richard Guy wrote:

> Start from the lighthouse theorem.

Even I am not quite sure what this is, so I'm sure that many
geometry-forum subscribers won't. But I think it's that if the
beams of three lighthouses all rotate at the same uniform rate,
then the triangle they form remains similar to itself, and rotates
at the same rate (while also expanding or contracting). Is this
right, Richard?

> This gives 9 potential Morley points for each pair of lighthice, BC,
> CA, AB. However, their (only 9 distinct) sides are (segments of) all 18
> Morley triangles.

... I've edited out quite a lot ...

> What I want to do is to show that the 3 sets of 9 lines from
> the lighthouse triangles are parallel (easy) and coincide in threes
> (not immediately obvious?) But I (have yet to) think (enough to
> show that) there is a proof for all 18 Morley triangles in one bang,
> and that Roussel's triangle is also in the picture.
>
> I meant to have thought this out more carefully, but found myself
> in a stream of (un?)consciousness. Can JHC make any sense of this?

I can't yet, but am certainly going to try to. I wasn't even
aware that the sides of the 18 Morley triangles coincide(d) in sets
of 3 (if I understand you correctly). You have the advantage over
me in having a picture that shows them all.

Let me say that there are in fact 18 Roussel triangles, not just
one. I don't know to what extent their edges or vertices coincide
(if at all).

After disproving my "general parallelism" conjecture, I made the
new one that (again replacing the trisectors by arbitrary isogonal
pairs of lines to the vertices) the Roussel-type triangle would always
be similar to the Morley-type one; but this too has failed (if my
calculations are correct). Nevertheless I'm sure that there is
something new and interesting to be found here, and I'm still thinking!

John Conway

Date Subject Author
9/12/98 Den Roussel
9/12/98 John Conway
9/13/98 Larry Cusick
10/27/98 John Conway
9/13/98 steve sigur
9/14/98 John Conway
9/15/98 John Conway
9/15/98 Richard Guy
9/15/98 John Conway
9/15/98 Richard Guy
9/15/98 Richard Guy
9/16/98 Floor van Lamoen
9/16/98 John Conway
9/17/98 Floor van Lamoen
9/17/98 John Conway
9/17/98 Floor van Lamoen
9/17/98 Russell Towle
9/17/98 John Conway
9/17/98 Russell Towle
9/17/98 Douglas J. Zare
9/19/98 Russell Towle
9/20/98 John Conway
9/20/98 John Conway
9/18/98 Antreas P. Hatzipolakis
11/10/98 Den Roussel