The homogeneity-based approach is well recognized in both optimal control and robust finite-time stabilization. M. I. Zelikin was the first to investigate weighted homogeneity in optimal synthesis. This paper is a tribute to Professor Zelikin and his contribution to geometric optimal control. After reviewing the background of homogeneous control applications, we focus on the problem of finite-horizon linear quadratic regulator design. The problem is studied under certain assumptions on the dilation symmetry (homogeneity) of the system. We show that the corresponding optimal solution can be obtained by solving an algebraic Riccati equation, similarly to the infinite-horizon case. The theoretical results are supported by numerical simulations.