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 K , is defined and shown, along with K - 1 , to play a natural and far-reaching role in relationships among triangles centers and loci in the plane of ABC .