Given $\triangle ABC, A(a,b), B(0,0),C(c,0)$. The locus is the line segment $AT$ on the line $x[(a-c)^{2}+b^{2}]+2b(a-c)y=a(a^{2}+b^{2}+c^{2})-2c(a^{2}+b^{2})$, point $T$ is the crossing point of this line with the perpendicular bisector of $AC$.