Hi Wilson,

Good to hear from you and to be back in triangle geometry after an

extended trip into the 4th dimension.

This is very interesting. At first I thought your result was wrong,

but after some thought I think it is right.

Here are my investigations that latch into your results:

I had suspected that strong points on weak lines were rare, but what

interests me is that they almost always happen for what I call "the

classical weak points." These are the inceter Io, Spieker So,

Feuerbach Fo, Nagel No, Gergonne Go, Mittenpunkt Mo, reflection of Io

in O = oIo, and some others. I sometimes call these the "friends of

the incenter" since they all depend on the incenter and/or incircles.

These points (other than oIo) are all part of what I call the "sweep

of the incenter."

Connect most (perhaps all) of these and you will find a strong point

on the line. I am making a chart of these strong points on weak lines

linked in "Steve's Voodoo pages" on my web site.

For fundamental reasons I want to add the point :1/(c-a): = isotomic

of infinity point on the dual of Io (the dual = tripolar of the

isotomic conjugate). Comment: In the projective world the tripolar is

nicer, but in the affine world the dual is nicer.

This point seems to me to be and extremely important point. It is to

the weak points as the Steiner point is to the strong points.

But...join it to the other friends of the incenter and you do not

seem to get lines with strong points as often.

This now connects me to your result. Here is what I have been

thinking: that points whose coordinate have a factor of (c-a) behave

differently than points that do not. I first notice this in some work

by Darij and me on the Shiffler and coSpieker points (linked on my

website).

Now almost all the lines between classical weak points have this

factor. For example the weak line G--Io = : c-a : which is the

tripolar of the point :1/(c-a): .

So my topic for investigation, based on your results, is: are points

like :1/(c-a): and its extraversions on circumconics, and, if so,

which ones.

This could be fun since it combines all the threads of thought in my

head at the moment.

Friendly from Atlanta,

Steve

On Feb 18, 2006, at 3:48 AM, Wilson Stothers wrote:

> After the last two posts, we have a theorem - with hindsight, it's

> rather obvious!

>

> Suppose that P is a weak point. Then L, the tripolar of P, contains

> a strong point Q if and only if P and all its extraversions lie on a

> circumconic C. In such a case, Q is the perspector of C.

> Proof

> Observe that a point Q is on the tripolar of P if and only if P is

> on the circumconic with perspector Q. Now Q will be strong if and

> only if it lies on the tripolars of all the extraversions of P. This

> requires that P is weak so that we have at least two tripolars.

> Thus, L contains a strong point if and only if the extraversions lie

> on a circumconic C. The strong point is then the perspector of C.

>

> Clearly, for a point P with more than two versions, it is unlikely

> that all versions lie on a single circumconic. Well, Steve, it now

> seems that the early evidence is misleading - weak lines will seldom

> contain strong points!

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