A triangle center associated with the Apollonius circle of the excircles of a triangle
Let E_A, E_B, E_C be the excircles of a given triangle ABC. Let S be the Apollonius circle internally tangent to the three excircles. Denote the point of tangency of E_A with S by T_A. Define T_B and T_C similarly. Draw the circle S_A that is tangent to AB and CA, and internally tangent to S, other than the three excircles. Denote the point of tangency of S_A with S by U_A. Define S_B, S_C, U_B, and U_C cyclically. Are the three lines T_A U_A, T_B U_B, T_C U_C concurrent? Move around A, B, and C to see if the proposition is true or not.