Fig. 29
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 [pg 66] 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.