Given the circle, midpoint $O(0,0)$ and radius $r$; points $B(0,r)$ and $C(c,w)$ are fixed, $w^{2}+c^{2}=r^{2}$. Variable point $A(\lambda,v)$ with $\lambda^{2}+v^{2}=r^{2}$. Point $M(\frac{\lambda+c}{2},\frac{v+w}{2})$. The line $BF\ :\ y=\frac{c-\lambda}{v-w}(x-r)$. The line $ME\ :\ y=\frac{cv-\lambda w}{c\lambda-r^{2}+vw}(x-c)$. The locus of the intersection points of these lines is half of the circle, going through $B$, $x^{2}+y^{2}-(c+r)x-\frac{w(r-c)}{c}y+cr=0$.