Несподіваний ефект формул Ейлера

This paper explores significant aspects of geometry, specifically problems and theorems related to two classical Euler's formulas: 〖IO〗^2=R^2-2Rr, which describes the distance between the centers of the circumcircle and the incircle, and 〖(OI_a)〗^2=R^2+2Rr_a, which characterizes the distance between the centers of the circumcircle and the A-excircle. The author notes that, despite their importance, one of the problems proposed by S. I. Zettel in his book Problems on Maxima and Minima has been largely overlooked in the mathematical community. The paper demonstrates how the application of formulas for the radii of the incircle and excircle, r_a=4Rsin A/2 cos B/2 cos C/2 and r=4Rsin A/2 sin B/2 sin C/2, not only simplifies the solution of the problem but also leads to a new extension of this result. The key idea is the use of the "analogy" method, which allows the discovery of new relationships and makes this approach appealing and useful for a broad range of mathematical researchers. Additionally, the paper includes a discussion of theorems and lemmas that will be applied in the proofs of the results, with the expectation that the material will be practically useful for readers.