Let $\{O_A,O_B,O_C\}$ be the Circumcenters of $\Delta BPC,\Delta APC,\Delta APB$ respectively. So, $\{\Delta AO_CB,AO_BC,\Delta BO_AC\}$ are Similar Isosceles Triangles erected on sidelines $AB,AC,BC$ respectively. So by Jacobi $\{AO_A,BO_B,CO_C\}$ are Concurrent $\implies P\in$ Neuberg Cubic of $\Delta ABC\implies$ Euler Lines of $\{\Delta APB,\Delta BPC,\Delta CPA,\Delta ABC\}$ are concurrent. $\blacksquare$