[Figure 27] 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_. *103. Pole and polar projectively related.* Another very important theorem comes directly from Fig. 26. _As a point __A__ moves along a straight line its polar with respect to a conic revolves about a fixed point and describes a pencil projective to the point-row described by __A__._ For, fix the points _L_ and _N_ and let the point _A_ move along the line _AQ_; then the point-row _A_ is projective to the pencil _LK_, and since _K_ moves along the conic, the pencil _LK_ is projective to the pencil _NK_, which in turn is projective to the point-row _C_, which, finally, is projective to the pencil _OC_, which is the polar of _A_.