Poles and polars have several useful properties.

If a point P lies on a line l, then the pole L of the line l lies on the polar p of point P.

If a point P moves along a line l, its polar p rotates about the pole L of the line l.

If two tangent lines can be drawn from a pole to the conic section, then its polar passes through both tangent points.

If a point lies on the conic section, its polar is the tangent through this point to the conic section.

If a point P lies on its own polar line, then P is on the conic section.

Each line has, with respect to a non-degenerated conic section, exactly one pole.

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