I've been stuck with this problem: An ellipse can be defined as 1) locus of points for which is constant the sum of the distances from two fixed points (foci) 2) locus of points for which is constant the ratio between the distances from a fixed point (focus) and a fixed line (directrix). The ratio value is the eccentricity of the ellipse. I would like to find a pure geometric demonstration that property 1) implies 2) and/or vice-versa. It's easy to demonstrate the equivalence between the two definitions if one translates the properties into analytical relationships between the coordinates of the locus points (cartesian or polar). I'm not interested in that. I would like to find a (possibly elegant) proof of the equivalence between definition 1) and 2) via pure geometric arguments (a la Euclide): with no coordinates but just geometric relationships, based on ruler and compass. I've searched the net and various textbook but haven't find nothing similar (even if I have found out several interesting geometrical constructions based on properties 1) or 2) that could be the first half of the bridge to proving the equivalence, i.e. the Boscovich circle seems interesting). Any help from a true geometry lover will be appreciated.