The isoscelizer in a triangle cuts off an isosceles triangle from an angle region of the original one. There is a single triad of three isoscelizers with the property that they give rise to four congruent "small" triangles in the inside of the "big" triangle. Among the four triangles, the central one is called the central Yff triangle. The proofs that I have seen start from the big triangle; employ the concept of similarity; and arrive at four small congruent triangles. I found it simpler to start from any "small" triangle, construct three more congruent triangles, and build up the "big" triangle (see first picture attached). This construction only needs the concept of congruence, so it applies to absolute geometry (plane, spherical and hyperbolic geometry). It is also easy to prove the existence of three tangent circles in the inside of the triangle. The dual construction on the sphere gives four small triangles that reminds me of the four congruent triangles created by the three midlines in Euclidean geometry; but on the sphere we get three more isosceles triangles, still in the inside of the original triangle (second picture).
I take the opportunity to wish good health and undiminshed energy to Peter Yff beyond his ninetieth birthday!
Thank you very much in advance. Did you also consider non-Euclidean cases?
Dear Peter, I apologize for responding so late for two reasons: End-of-semester rush days; even more, I wanted to work a bit more on the non-Euclidean cases. If I have a breather, I will send more. For your kind offer, I would love to receive your paper to my address: istván Lénárt, H-1145 Amerikai út 40/B., Budapest, Hungary. Thank you very much.
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