In both of the views at the top of the “conics3” page, we have

• a square of tangents to a circle (which, being a planar section of the indicated cone, is a conic), and
• a square of chords formed by joining the contact-points of the tangents.

The diagonals of both squares have the same point in common. It is the center of the circle.

Each diagonal of the chord-square lies parallel to a pair of opposite sides of the tangent-square, and vice versa, and so, since there are four such diagonals in all, we have four sets of three parallel lines, each set with its own common point at infinity in the line at infinity of the cutting-plane.

• Each “corner” of the tangent-square is the dual and pole of a side of the chord-square, so that the side is the polar of the corner.
• Each “corner” of the chord-square is the dual and pole of a side of the tangent-square, so that the side is the polar of the corner.

F is the pole of line f,
Line f is the polar of F.

B is the pole of line b,
Line b is the polar of B.

It follows that

• the chord-square is the dual, and in fact polar, version of the tangent-square
• the four points in the line at infinity stand in involution, in conjugate pairs
• the circle's center, as the common point of the four diagonals, is the pole of the line at infinity
• the line at infinity cuts a circle in a pair of imaginary points
• all circles on a plane have the line at infinity of that plane as their common polar
• all circles on a plane have the same two imaginaries in the line at infinity in common