Given the regular pentagon $ABCDE\ :\ A(0,0),B(1,0), /$, midpoint $M(\frac{1}{2},\frac{1}{2}\tan 54^{\circ})$. Choose the point $P(\frac{1}{2},\lambda)$. Circumcircle $x^{2}+y^{2}-x+\frac{1-4\lambda^{2}}{4\lambda} \cdot y=0$. The line $AE\ :\ t=-\tan 72^{\circ} \cdot x$ intersects this circle in the point $Q(\cos^{2}72^{\circ}+\frac{1-4\lambda^{2}}{4\lambda}\sin 72^{\circ}\cos 72^{\circ})$. The perpendicular $y-\lambda=\tan 54^{\circ}(x-\frac{1}{2})$ intersects this circle in the point $R(\frac{1}{2}-\frac{4\lambda^{2}+1}{4\lambda}\sin 54^{\circ}\cos 54^{\circ},\lambda-\frac{4\lambda^{2}+1}{4\lambda}\sin^{2}54^{\circ})$. The circle, midpoint $R$ and radius $AR$, goes through $Q$.