If one point lies on the polar of a second, then the second lies on the polar of the first.
100. Conjugate points and lines. Such a pair of points are said to be conjugate with respect to the conic. Similarly, lines are said to be conjugate to each other with respect to the conic if one, and consequently each, passes through the pole of the other.
Fig. 27
101. Construction of the polar line of a given point. Given a point P, if it is within the conic (that is, if no tangents may be drawn from P to the conic), we may construct its polar line by drawing through it any two chords and joining the two points of intersection of the two pairs of tangents at their extremities. If the point P is outside the conic, we may draw the two tangents and construct the chord of contact (Fig. 27).
102. Self-polar triangle. In Fig. 26 it is not difficult to see that AOC is a self-polar triangle, that is, each vertex is the pole of the opposite side. For B, M, O, K are four harmonic points, and they project to C in four harmonic rays. The line CO, therefore, meets the line AMN in a point on the polar of A, being separated from A harmonically by the points M and N. Similarly, the line CO meets KL in a point on the polar of A, and therefore CO is the polar of A. Similarly, OA is the polar of C, and therefore O is the pole of AC.