conics

These had already been proved in earlier treatises, which Archimedes refers to as the “Elements of Conics”.

Two conics which have four common tangents have always one and only one common polar-triangle.

Similarly, all conics touching four fixed lines form a system such that any fifth tangent determines one and only one conic.

The corresponding remark holds for the problem of drawing the conics which touch four lines and pass through a given point.

It will cut the two conics first at T, and therefore each at some other point which we call A and B respectively.