*117.* For the circumscribed quadrilateral, Brianchon’s theorem gave (§ 88) _The lines joining opposite vertices and the lines joining opposite points of contact are four lines meeting in a point._ Take now for two of the tangents the asymptotes, and let _AB_ and _CD_ be any other two (Fig. 29). If _B_ and _D_ are opposite vertices, and also _A_ and _C_, then _AC_ and _BD_ are parallel, and parallel to _PQ_, the line joining the points of contact of _AB_ and _CD_, for these are three of the four lines of the theorem just quoted. The fourth is the line at infinity which joins the point of contact of the asymptotes. It is thus seen that the triangles _ABC_ and _ADC_ are equivalent, and therefore the triangles _AOB_ and _COD_ are also. The tangent AB may be fixed, and the tangent _CD_ chosen arbitrarily; therefore _The triangle formed by any tangent to the hyperbola and the two asymptotes is of constant area._ *118. Equation of hyperbola referred to the asymptotes.* Draw through the point of contact _P_ of the tangent _AB_ two lines, one parallel to one asymptote and the other parallel to the other. One of these lines meets _OB_ at a distance _y_ from _O_, and the other meets _OA_ at a distance _x_ from _O_. Then, since _P_ is the middle point of _AB_, _x_ is one half of _OA_ and _y_ is one half of _OB_. The area of the parallelogram whose adjacent sides are _x_ and _y_ is one half the area of the triangle _AOB_, and therefore, by the preceding paragraph, is constant. This area is equal to _xy · __sin__ α_, where α is the constant angle between the asymptotes. It follows that the product _xy_ is constant, and since _x_ and _y_ are the oblique coördinates of the point _P_, the asymptotes being the axes of reference, we have _The equation of the hyperbola, referred to the asymptotes as axes, is __xy =__ constant._ This identifies the curve with the hyperbola as defined and discussed in works on analytic geometry. [Figure 30] FIG. 30 *119. Equation of parabola.* We have defined the parabola as a conic which is tangent to the line at infinity (§ 110). Draw now two tangents to the curve (Fig. 30), meeting in _A_, the points of contact being _B_ and _C_. These two tangents, together with the line at infinity, form a triangle circumscribed about the conic. Draw through _B_ a parallel to _AC_, and through _C_ a parallel to _AB_. If these meet in _D_, then _AD_ is a diameter. Let _AD_ meet the curve in _P_, and the chord _BC_ in _Q_. _P_ is then the middle point of _AQ_. Also, _Q_ is the middle point of the chord _BC_, and therefore the diameter _AD_ bisects all chords parallel to _BC_. In particular, _AD_ passes through _P_, the point of contact of the tangent drawn parallel to _BC_.