Given the circle, midpoint $O(0,0)$ and radius $1$. Points on this circle: $A(1,0),B(b,u),C(c,v),D(d,w)$ with $u=\sqrt{1-b^{2}}$, and so on. The tangent line in $A\ :\ x=1$ intersects the tangent line in $D\ :\ xd+yw=1$ in the focus $F_{1}(1,\frac{1-d}{w})$. The tangent line in $B\ :\ bx+uy=1$ intersects the tangent line in $C\ :\ cx+vy=1$ in the focus $F_{2}(\frac{v-u}{bv-cu},\frac{b-c}{bv-cu})$. The tangent line in $A\ :\ x=1$ intersects the tangent line in $C$ in the point $P(1,\frac{1-c}{v})$. The tangent line in $B$ intersects the tangent line in $D$ in the point $Q(\frac{w-u}{bw-du},\frac{b-d}{bw-du})$. To prove: $PF_{1}+PF_{2}=QPF_{1}+QF_{2}=\frac{u-v}{v(bv-cu)}+\frac{cw-v(d-1)}{vw}$.