§ 65. The second property of the polar or pole gives rise to the theorem—
| From a point in the plane of a conic, two, one or no tangents may be drawn to the conic, as its polar has two, one, or no points in common with the curve. | A line in the plane of a conic has two, one or no points in common with the conic, according as two, one or no tangents can be drawn from its pole to the conic. |
Of any point in the plane of a conic we say that it was without, on or within the curve according as two, one or no tangents to the curve pass through it. The points on the conic separate those within the conic from those without. That this is true for a circle is known from elementary geometry. That it also holds for other conics follows from the fact that every conic may be considered as the projection of a circle, which will be proved later on.
The fifth property of pole and polar stated in § 64 shows how to find the polar of any point and the pole of any line by aid of the straight-edge only. Practically it is often convenient to draw three secants through the pole, and to determine only one of the diagonal points for two of the four-points formed by pairs of these lines and the conic (fig. 22).
These constructions also solve the problem—
From a point without a conic, to draw the two tangents to the conic by aid of the straight-edge only.
For we need only draw the polar of the point in order to find the points of contact.
§ 66. The property of a polar-triangle may now be stated thus—
In a polar-triangle each side is the polar of the opposite vertex, and each vertex is the pole of the opposite side.
| Fig. 23. |