The figure speaks from itself. The central symmetry of the hyperbola and its asymptotics about its center O implies the equality of the red segments AE = BC = ... etc. and the equality of the blue segments DE = FC = ... etc. In other words, every line defines through the hyperbola and its asymptotics two segments having a common middle (f.e. EC, AB). This has an important consequence for recangular hyperbolas (see RectangularAsProp.html ).