Properties of the Kiss–Bíró configuration

In the plane of a triangle ABC, by using the circumcenter O and the orthocenter H, we can naturally construct certain triangles DEF and $$D'E'F'$$ that are similar to ABC. These triangles serve as a basis for the Kiss–Bíró configuration. Their circumcircles meet in the orthocenter, H, and also a remarkable second point, $$H'$$ , for which barycentric coordinates and properties are derived. A third triangle, $$D''E''F''$$ , is introduced and proved inversely similar to and orthologic to ABC, and other properties of this third triangle are given. The similarity transformation that maps $$D''E''F''$$ onto ABC, denoted by $${\mathcal {K}}$$ , is defined and shown, along with $${\mathcal {K}}^{-1}$$ , to play a natural and far-reaching role in relationships among triangles centers and loci in the plane of ABC.