Given the circle $x^{2}+y^{2}=r^{2}$ with two fixed points $A(-\frac{r}{\sqrt{2}},-\frac{r}{\sqrt{2}}),C(r,0)$. Points $B(r\cos \alpha,r\sin \alpha)$ and $D(r\cos \alpha,-r\sin \alpha)$. The lines $BD$ and $AC$ intersect in the point $M(r\cos \alpha,(\sqrt{2}-1)(r\cos \alpha -r)$. The locus of the circumcenter of $\triangle ABM$ is the arc of the circle, lying above the x-axis. The equation of this circle $x^{2}+y^{2}+rx+r(\sqrt{2}-1)y=0$, midpoint $(-\frac{r}{2},-\frac{(\sqrt{2}-1)r}{2})$, going through $A,(0,0),(-r,0)$.