Given $O(0,0)$; let $\angle (Op_{1},Op_{2})=2\alpha$. Choose midpoint $O_{1}(\lambda,0)$ and radius $r$, then $r=\lambda \sin 2\alpha$. Choose midpoint $O_{2}(t, t \cdot \tan \alpha)$ and radius $t \cdot \tan \alpha$. Then $(O_{1}O_{2})^{2}=t^{2}\tan^{2}\alpha+(\lambda-t)^{2}$, calculating $t=\lambda(1-\sqrt{2}\sin \alpha)^{2}$. The line $O_{1}O_{2}\ y=\frac{t \cdot \tan \alpha}{t-\lambda}(x-\lambda)$ cuts the circles in the point of tangency. The locus of this point is the line $y=\frac{\sin 2\alpha}{\cos 2\alpha+2} \cdot x$.