9. Given five tangents to a conic, to draw a tangent which shall be parallel to any one of them. 10. The lines _a_, _b_, _c_ are drawn parallel to each other. The lines _a’_, _b’_, _c’_ are also drawn parallel to each other. Show why the lines (_ab’_, _a’b_), (_bc’_, _b’c_), (_ca’_, _c’a_) meet in a point. (In problems 6 to 10 inclusive, parallel lines are to be drawn.) CHAPTER VI - POLES AND POLARS *95. Inscribed and circumscribed quadrilaterals.* The following theorems have been noted as special cases of Pascal’s and Brianchon’s theorems: _If a quadrilateral be inscribed in a conic, two pairs of opposite sides and the tangents at opposite vertices intersect in four points, all of which lie on a straight line._ _If a quadrilateral be circumscribed about a conic, the lines joining two pairs of opposite vertices and the lines joining two opposite points of contact are four lines which meet in a point._ [Figure 26] 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 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 _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.