Ars Mathematica: Solución al problema 61

\begin{matrix} {(\frac{a}{2} + r)}^{2} & = & {(\frac{a}{2} - r)}^{2} + (a - r)^{2} \\ {(\frac{a}{2} + r)}^{2} - {(\frac{a}{2} - r)}^{2} & = & (a - r)^{2} \\ (\frac{a}{2} + r + \frac{a}{2} - r) (\frac{a}{2} + r - \frac{a}{2} + r) & = & (a - r)^{2} \\ 2 a r & = & a^{2} - 2 a r + r^{2} \\ r^{2} - 4 a r + a^{2} & = & 0 \\ r & = & \frac{4 a \pm \sqrt{16 a^{2} - 4 a^{2}}}{2} \\ r & = & \frac{4 a \pm 2 a \sqrt{3}}{2} = a \frac{2 \pm \sqrt{3}}{2} \end{matrix}

r = a \frac{2 - \sqrt{3}}{2}

Ars Mathematica