The "ex" in extraversion

The incircle touches the edges of ABC in points that are perspective to ABC at the Gergonne point, which is thereby dependent on the incenter and the incircle for existence. So too for many others. These points are quartile in that they come in extra-versions, four to be precise via John Conway's extraversion symmetry. If we deleted the incenter and incircle from existence, just think of how much else would be deleted. Kapetis has a very impressive sequence in volume 1 of his great books that begins with the incenter and its tripolar and derive point after point. How much of the Encyclopedia of Triangle Centers would go away if we could delete the incenter from existence?

So I have been thinking about this and have thought of a way to explicitly derive quartile points from the four incenters via affine transformations, a part of my affine project.

Let x_a be the projective operation that takes ABCP with P ~ ( l:m:n ) into ABCA^P where A^P ~ ( –l:m:n ) is the A-ex-version of P, the A-vertex of the preCevian triangle of P. Similarly for x_b and x_c. Note that x_a x_b = x_c in the projective plane. These operations satisfy the Kein group where "o" is the identity.

This group is familiar as part of the group defined on pivotal cubics and for being the desmic triple muliplication law.

Now consider the quadrangle { P, x_aP, x_bP, x_cP} of P and its ex points and operate with any of the affine invariant operations in this chart, which shows the automatic, affine invariant generation of triangle points.

In this chart, the horizontal lines represent the operation t, the isotomic conjugate. The oblique lines represent m, the medial operation, and d, its inverse. These relationships are maintained under affine trasformations.

In this chart we showed the operations to P alone, but now let us consider the operations applied to the quadrangle of P and its ex-points. We will use dP :y: -> :z+x–y: as an example.

d applied to each of {{l, m, n}, {–l, m, n}, {l, –m, n}, {l, m, –n}} yields

{{–l + m + n, l – m + n, l + m – n},
{l + m + n, –l – m + n, –l + m – n},
{–l – m + n, l + m + n, l – m – n},
{–l + m – n, l – m – n, l + m + n}}

 

	{–l + m + n, l – m + n, l + m – n}	{l + m + n, –l – m + n, –l + m – n}	{–l – m + n, l + m + n, l – m – n}	{–l + m – n, l – m – n, l + m + n}
	{1/(–l + m + n) , 1/( l – m + n) , 1/(l + m – n)}		{1/(–l + m + n) , (l+m+n)/(– l + m + n)(–l + m – n) , 1/(l + m – n)}
	: ll – mm + nn :	A	B	C

 

Which appears to be one quadrangle of a Desmic triple of quadrangle. Now if P were the incenter, these four points would be the extraversions of N_o, the Nagel point. The rub is that these were derived from purely affine transformations of the incenter, which is not like the extraversion symmetry at all.

With a little thought, one can see that the above process generates essentially all of the weak points and their extraversions.

The following chart shows how this works if P = incenter. Each circle is to be understood at standing for the point and its extraversions.