91. Harmonic tangents. We have seen that a variable tangent cuts out on any two fixed tangents projective point-rows. It follows that if four tangents cut a fifth in four harmonic points, they must cut every tangent in four harmonic points. It is possible, therefore, to make the following definition:
Four tangents to a conic are said to be harmonic when they meet every other tangent in four harmonic points.
92. Projectivity and perspectivity. This definition suggests the possibility of defining a projective correspondence between the elements of a pencil of rays of the second order and the elements of any form heretofore discussed. In particular, the points on a tangent are said to be perspectively related to the tangents of a conic when each point lies on the tangent which corresponds to it. These notions are of importance in the higher developments of the subject.
Fig. 25
93. Brianchon's theorem may also be applied to a degenerate conic made up of two points and the lines through them. Thus(Fig. 25),
If a, b, c are three lines through a point S, and a', b', c' are three lines through another point S', then the lines l = (ab', a'b), m = (bc', b'c), and n = (ca', c'a) all meet in a point.