engleski

Centers of the golden ratio Archimedean twin circles

We explore some properties of the geometric configuration when arbelos of the same ratio are constructed on sides of a triangle. The centers of the Archimedean twin circles of these arbelos determine two triangles that are either orthologic or homologic to the base triangle only when the common ratio of arbelos is the number related to the golden ratio. We also consider several triangles associated to the base triangle and build arbelos of the same ratio on their sides and seek when their centers of the Archimedean twin circles give triangles that are either orthologic or homologic to the base triangle. When we construct arbelos on sides of pedal and antipedal triangles of points analogous statements are possible only for points on the Brocard axis and on the Kiepert hyperbola of the base triangle.

arbelos; Archimedean twin circles; orthology; homology; Napoleon triangles; Torricelli triangles; pedal triangle; antipedal triangle; golden ratio

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