There are some methods to construct a conic, example: Based on Pascal theorem, Steiner construction, .....I propose a method to construct a conic as follows:

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

 

My question: I am looking for a proof that locus of $H, G$ is a conic section when $E$ be moved on line $L_1$.

See also:

• A chain of six circles associated with a conic

There are some methods to construct a conic, example: Based on Pascal theorem, Steiner construction, .....I propose a method to construct a conic as follows:

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

My question: I am looking for a proof that locus of $H, G$ is a conic section when $E$ be moved on line $L_1$.

See also:

• A chain of six circles associated with a conic

There are some methods to construct a conic, example: Based on Pascal theorem, Steiner construction, .....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

See also:

• A chain of six circles associated with a conic

There are some methods to construct a conic, example: Based on Pascal theorem, Steiner construction, .....I propose a method to construct a conic as follows:

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

My question: I am looking for a proof that locus of $H, G$ is a conic section when $E$ be moved on line $L_1$.

See also:

• A chain of six circles associated with a conic