Significance of homothety mapping incircle to circumcircle

Question

Are there any special properties of the homothety mapping the incircle of a triangle to the circumcircle? For example are the centers of this homothety triangle centers?

Answer 1

Triangle centers are points whose relative position (to the vertices) does not change under similarity transformations applied to triangles. The center of homothety transforming the incircle to the circumcircle is such a triangle center.

One can demonstarate this statement based on the following observations:

1. Translation, rotation, and reflection will not change the relative position of the this center. This is, hopefully, obvious.
2. It is, then, enough to show that dilatation does not change the relative position of this center either.

Proof: Notice that when applying a dilatation to a triangle the incircle and the circumcircle will not have to be reconstructed. The same dilatation will transform both as shown in the following figure.

Also, the structure of the homothety transforming the incircle to the circumcircle (the yellow solid triangle and the double circled center) will be transformed by the homothety transforming the triangles. (See the broken yellow lines.) Consider that a homothety is determined by its action on two parallel segments.

(Note that the two radii (blue-red) determining the yellow center can be chosen arbitrarily.)