Fig. 26
96. Definition of the polar line of a point. Consider the quadrilateral K, L, M, N inscribed in the conic (Fig. 26). It determines the four harmonic points A, B, C, D which project from N in to the four harmonic points M, B, K, O. Now the tangents at K and M meet in P, a point on the line AB. The line AB is thus determined entirely by [pg 57] the point O. For if we draw any line through it, meeting the conic in K and M, and construct the harmonic conjugate B of O with respect to K and M, and also the two tangents at K and M which meet in the point P, then BP is the line in question. It thus appears that the line LON may be any line whatever through O; and since D, L, O, N are four harmonic points, we may describe the line AB as the locus of points which are harmonic conjugates of O with respect to the two points where any line through O meets the curve.
97. Furthermore, since the tangents at L and N meet on this same line, it appears as the locus of intersections of pairs of tangents drawn at the extremities of chords through O.
98. This important line, which is completely determined by the point O, is called the polar of O with respect to the conic; and the point O is called the pole of the line with respect to the conic.
99. If a point B is on the polar of O, then it is harmonically conjugate to O with respect to the two intersections K and M of the line BC with the conic. But for the same reason O is on the polar of B. We have, then, the fundamental theorem