I'll bring you a paper to prove the equivalence of two definitions of ellipse by pure geometry. Finally, the proof diagram of hyperbola is given, which shows that hyperbola can be proved by simulating ellipse. But I think we should use the semi-major axis length A, the semi-focal length C, the eccentricity E and the directrix A? /c I don't know if it meets your requirements, but you can refer to it.