The incenter project

The more symmetry the more importance – JHC

Two points have a midpoint. Extending this sense of center to the triangle leads to the centroid and isotomic conjugacy.

Two lines have a midline. Extending this sense of center to the triangle leads to the incenter and isogonal conjugacy.

But while the centroid is affinely invariant, the incenter is neither projectively, affinely, nor inversively invariant. The puzzle is that much of the structure of triangle geometry comes from this point that contains none of the symmetries of the plane itself. This was my great puzzle for a long time, but it is resolved now.

The incenter is the generator of affine structure in the plane. This is both why the incenter is so important and affine geometry so effective.

A most noticible fact for the incenter is that there are 4 of them. Virtually all quartile, weak points originate with the quartile nature of the incenter.

The incentral quadrangle is both its own ex and extra-versions.

Here are a series of papers that try to puzzle this out and explore the importance of the incenter.

	Friends of the Incenter
	The orbit of the incenter; a striking pattern in a large group of triangle points, it is also invariant under the projective transformation that takes ABCG to ABCIo
	The incentral triangle; this triangle generates so many surprising structures, it must be very important.
	The "ex" in extraversion, thoughts on the incenter.
	Weak Conics