Euclidean geometric description of radial centers of a triangle

We consider the convolution of a radially symmetric function and the characteristic function of a triangle. We describe the location of a critical point of the convolution in the Euclidean geometric manner. The description slightly shortens the proof of the analytic characterizations of equilateral triangles shown in [S. Sakata, Ann. Mat. Pura Appl. 200 (2021)].