Every polar-triangle has one and only one vertex within the conic.
We add, without a proof, the theorem—
The four points in which a conic is cut by two conjugate polars are four harmonic points in the conic.
§ 68. If two conics intersect in four points (they cannot have more points in common, § 52), there exists one and only one four-point which is inscribed in both, and therefore one polar-triangle common to both.
Theorem.—Two conics which intersect in four points have always one and only one common polar-triangle; and reciprocally,
Two conics which have four common tangents have always one and only one common polar-triangle.
Diameters and Axes of Conics
§ 69. Diameters.—The theorems about the harmonic properties of poles and polars contain, as special cases, a number of important metrical properties of conics. These are obtained if either the pole or the polar is moved to infinity,—it being remembered that the harmonic conjugate to a point at infinity, with regard to two points A, B, is the middle point of the segment AB. The most important properties are stated in the following theorems:—
The middle points of parallel chords of a conic lie in a line—viz. on the polar to the point at infinity on the parallel chords.
This line is called a diameter.