The affine project
I began looking at affine geometry several years ago just to rapidly get familiar with it and move on to the more important (I thought) inversive geometry. Again and again affine geometry seemed the easy path to many results so I kept looking at it. The great realization is that the incenter contains the pattern that generates the affine structure of the plane. The incenter is so important to triangle geometry so that affine theory and triangle geometry are surprisingly linked throught the incenter.
There is an affine background of points (those with constant barycentric coordinates) that maintain the same relationship to reference triangle ABC. I tend to think that many properties of points in triangle geometry are really properties of these points. For example the point (2:3:4) is both the perspector of an inconic and a cricumconic to ABC. For some triangle the orthocenter, say, might have these coordinates. In then "borrows" the conic of the point on which it resides until it moves over another point, whereby it borrows the properties of that point.
Here are a series of papers that expound this point of view.
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