There are some methods to construct a conic, example: [Based on Pascal theorem](https://en.wikipedia.org/wiki/Five_points_determine_a_conic#Construction), [Steiner construction](https://en.wikipedia.org/wiki/Conic_section#Steiner's_projective_conic_definition), .....I propose a method to construct a conic as follows:

> Let two parallel lines $L_1 \parallel L_2$, let $A, B, C, D$ be  four points in the plane. Let $L$ be a line on the plane such that $L \parallel AB$. Let points $E=L \cap L_1, F=L \cap L_2$. Let circle (center $E$, radius $ED$) meets the circle (center $F$, radius $FC$) at two points $H$, $G$. 

> **My question:** I am looking for a proof that the locus of $G$, $H$ is a conic when we moved $L$ on the plane

[![enter image description here][1]][1]

See also:

* [A chain of six circles associated with a conic](https://mathoverflow.net/questions/234722)


  [1]: https://i.sstatic.net/FXwMw.png