Circumscribed hyperbolic triangles (Q2673900)

The subject of this short paper is the equivalence condition under which the hyperbolic triangle has a circumscribed circle. Note that a hyperbolic circle in the unit disc model is a Euclidean circle. We have a unique Euclidean circle \(C\) passing through three distinct points in the unit disc model, and we can consider whether it is wholly contained in the unit disc. (If the Euclidean circle \(C\) is tangent to the unit disc, it is a holocycle.) The main conclusion of this paper is that the existence of a circumscribed circle of a hyperbolic triangle is equivalent to an inequality similar to the triangle inequality for \(\sinh\) of the edge lengths of the hyperbolic triangle. The proof is straightforward, but the phenomenon is interesting. From there, the author also mentions the condition of hyperbolic triangles inscribed in a holocycle and the triangle inequalities for polygons inscribed in hyperbolic circles. As mentioned in the last section, we more easily understand equality in the case of holocycle in the upper-half model. A polygon inscribed in a holocycle is a sequence of points with the same height. Thus equality of summation holds. Discussing the ``mathematical novelty'' of these elementary geometry papers would be one topic of discussion. The reviewer wants young researchers to refrain from mass-producing such simple papers in order to earn achievements. On the other hand, the reviewer believes that elementary geometry, even hyperbolic geometry, is open to all mathematicians. Therefore considering `new theorems' (this means propositions we don't have in standard textbooks) is a ``meaningful mathematical activity.'' The existence of such papers, especially by such an eminent author, as an archive is so welcome.