An Evolutionary Structure of Convex Quadrilaterals

We solve the problem of the evolution of convex quadrilaterals by applying the inverse weighted Fermat-Torricelli problem, the invariance property of the weighted Fermat-Torricelli point in the plane $\mathbb{R}^2$, two-dimensional sphere $S^2$ and the two-dimensional hyperboloid $H^2$. This means that the property of plasticity is inherited by some evolutionary convex quadrilaterals. An important application is the connection of the Fermat-Torricelli point with the fundamental equation of P. de Fermat.