If an equilateral pentagon is inscribed in a circle which has a rational diameter then the side of the pentagon is that irrational (straight line) called minor. * For let the equilateral pentagon $ABCDE$ have been inscribed in the circle $ABCDE$ which has a rational diameter. * I say that the side of pentagon [$ABCDE$] is that irrational (straight line) called minor.
If the circle has unit radius then the side of the pentagon is \[\frac{\sqrt{10-2\,\sqrt{5}}}2.\]
However, this length can be written in the minor. \[\sqrt{5}\sqrt{\left(1+\frac{\rho}{\sqrt{1+\rho^2}}\right)\frac 12} - \sqrt{\left(1-\frac{\rho}{\sqrt{1+\rho^2}}\right)\frac 12}, \]
with $\rho=2$.
Proofs: 1
Proofs: 1
