_AP · BQ = ± b__2__._ [Figure 31] FIG. 31 Since _AD_ and _BC_ are parallel tangents, _PQ_ is a diameter and the conjugate diameter is parallel to _AD_. The middle point of _PQ_ is the center of the conic. We take now for the axis of abscissas the diameter _PQ_, and the conjugate diameter for the axis of ordinates. Join _A_ to _Q_ and _B_ to _P_ and draw a line through _S_ parallel to the axis of ordinates. These three lines all meet in a point _N_, because _AP_, _BQ_, and _AB_ form a triangle circumscribed to the conic. Let _NS_ meet _PQ_ in _M_. Then, from the properties of the circumscribed triangle (§ 89), _M_, _N_, _S_, and the point at infinity on _NS_ are four harmonic points, and therefore _N_ is the middle point of _MS_. If the coördinates of _S_ are _(x, y)_, so that _OM_ is _x_ and _MS_ is _y_, then _MN = y/2_. Now from the similar triangles _PMN_ and _PQB_ we have _BQ : PQ = NM : PM,_ and from the similar triangles _PQA_ and _MQN_, _AP : PQ = MN : MQ,_ whence, multiplying, we have _±b__2__/4 a__2__ = y__2__/4 (a + x)(a - x),_ where [formula] or, simplifying, [formula] which is the equation of an ellipse when _b__2_ has a positive sign, and of a hyperbola when _b__2_ has a negative sign. We have thus identified point-rows of the second order with the curves given by equations of the second degree. PROBLEMS 1. Draw a chord of a given conic which shall be bisected by a given point _P_. 2. Show that all chords of a given conic that are bisected by a given chord are tangent to a parabola. 3. Construct a parabola, given two tangents with their points of contact. 4. Construct a parabola, given three points and the direction of the diameters. 5. A line _u’_ is drawn through the pole _U_ of a line _u_ and at right angles to _u_. The line _u_ revolves about a point _P_. Show that the line _u’_ is tangent to a parabola. (The lines _u_ and _u’_ are called normal conjugates.) 6. Given a circle and its center _O_, to draw a line through a given point _P_ parallel to a given line _q_. Prove the following construction: Let _p_ be the polar of _P_, _Q_ the pole of _q_, and _A_ the intersection of _p_ with _OQ_. The polar of _A_ is the desired line. CHAPTER VIII - INVOLUTION [Figure 32] FIG. 32 *121. Fundamental theorem.* The important theorem concerning two complete quadrangles (§ 26), upon which the theory of four harmonic points was based, can easily be extended to the case where the four lines _KL_, _K’L’_, _MN_, _M’N’_ do not all meet in the same point _A_, and the more general theorem that results may also be made the basis of a theory no less important, which has to do with six points on a line. The theorem is as follows: