Conway established the following geometric fact: If the sides $ AB $ and $ AC $ of a triangle $ ABC $ are prolonged beyond the point $ A $ by the length of the oppositeside $ BC $ and the same is done with the vertices $ B $ and $ C $ , then the so-constructed 6 pointslie on the sole circle whose center coincides with the center of the inscribed circle.V.A. Alexandrov found a spatial analog of the Conway circle. Namely, if in a tetrahedron $ ABCD $ we mark three points on the prolongations of the edges $ AB $ , $ AC $ , and $ AD $ beyond the vertex $ A $ at distance from $ A $ to the half-perimeter of the opposite face $ BCD $ and then dothe same with the remaining vertices $ B $ , $ C $ , and $ D $ then the so-constructed 12 points lieon the same sphere if and only if $ ABCD $ is a frame tetrahedron.We address the multidimensional version of the fact for a simplex in the Euclidean space $ E_{n} $ .
