Heilbronn's problem of eight points in the square (Q6076812)

For any eight points \(P_1,P_2,\dots,P_8\) in the square, the smallest area of the \(\binom{8}{3}=56\) triangles formed by the eight points is less than or equal to \((\sqrt{13}-1)/36\), and the equality holds if and only if the configuration is congruent to the following eight points \[ \big(0,0\big),\quad\Big(\frac{1+\sqrt{13}}{6},0\Big),\quad\Big(1,\frac{7-\sqrt{13}}{18}\Big),\quad\big(1,1\big), \] \[ \Big(\frac{5-\sqrt{13}}{6},1\Big), \quad\Big(0,\frac{11+\sqrt{13}}{18}\Big), \quad\Big(\frac{5-\sqrt{13}}{6},\frac{7-\sqrt{13}}{9}\Big), \quad\Big(\frac{1+\sqrt{13}}{6},\frac{2+\sqrt{13}}{9}\Big). \] Symbolic computations are employed to reduce the problem into a nonlinear programming problem and find its optimal solution. All computations are done using Maple.