MathNet 2025 — 50 problems with figures (EN–VI) Question 9

A right prism $ABC A_1 B_1 C_1$ is given. It is known that triangles $A_1 BC$, $AB_1 C$, $ABC_1$, and $ABC$ are acute-angled. Prove that the orthocenters of these triangles, and the centroid of $ABC$ lie on a sphere.