Draw now another tangent, meeting _AB_ in _B’_ and _AC_ in _C’_. Then these three, with the line at infinity, make a circumscribed quadrilateral. But, by Brianchon’s theorem applied to a quadrilateral (§ 88), it appears that a parallel to _AC_ through _B’_, a parallel to _AB_ through _C’_, and the line _BC_ meet in a point _D’_. Also, from the similar triangles _BB’D’_ and _BAC_ we have, for all positions of the tangent line _B’C_, _B’D’ : BB’ = AC : AB,_ or, since _B’D’ = AC’_, _AC’: BB’ = AC:AB =_ constant. If another tangent meet _AB_ in _B"_ and _AC_ in _C"_, we have _ AC’ : BB’ = AC" : BB", _ and by subtraction we get _C’C" : B’B" =_ constant; whence _The segments cut off on any two tangents to a parabola by a variable tangent are proportional._ If now we take the tangent _B’C’_ as axis of ordinates, and the diameter through the point of contact _O_ as axis of abscissas, calling the coordinates of _B(x, y)_ and of _C(x’, y’)_, then, from the similar triangles _BMD’_ and we have _y : y’ = BD’ : D’C = BB’ : AB’._ Also _y : y’ = B’D’ : C’C = AC’ : C’C._ If now a line is drawn through _A_ parallel to a diameter, meeting the axis of ordinates in _K_, we have _AK : OQ’ = AC’ : CC’ = y : y’,_ and _OM : AK = BB’ : AB’ = y : y’,_ and, by multiplication, _OM : OQ’ = y__2__ : y’__2__,_ or _x : x’ = y__2__ : y’__2__;_ whence _The abscissas of two points on a parabola are to each other as the squares of the corresponding coördinates, a diameter and the tangent to the curve at the extremity of the diameter being the axes of reference._ The last equation may be written _y__2__ = 2px,_ where _2p_ stands for _y’__2__ : x’_. The parabola is thus identified with the curve of the same name studied in treatises on analytic geometry. *120. Equation of central conics referred to conjugate diameters.* Consider now a _central conic_, that is, one which is not a parabola and the center of which is therefore at a finite distance. Draw any four tangents to it, two of which are parallel (Fig. 31). Let the parallel tangents meet one of the other tangents in _A_ and _B_ and the other in _C_ and _D_, and let _P_ and _Q_ be the points of contact of the parallel tangents _R_ and _S_ of the others. Then _AC_, _BD_, _PQ_, and _RS_ all meet in a point _W_ (§ 88). From the figure, _PW : WQ = AP : QC = PD : BQ,_ or _AP · BQ = PD · QC._ If now _DC_ is a fixed tangent and _AB_ a variable one, we have from this equation _AP · BQ = __constant._ This constant will be positive or negative according as _PA_ and _BQ_ are measured in the same or in opposite directions. Accordingly we write