88. Circumscribed quadrilateral. If two pairs of lines in Brianchon's hexagon coalesce, we have a theorem concerning a quadrilateral circumscribed about a conic. It is easily found to be (Fig. 23)
The four lines joining the two opposite pairs of vertices and the two opposite points of contact of a quadrilateral circumscribed about a conic all meet in a point. The consequences of this theorem will be deduced later.
Fig. 24
89. Circumscribed triangle. The hexagon may further degenerate into a triangle, giving the theorem (Fig. 24) The lines joining the vertices to the points of contact of the opposite sides of a triangle circumscribed about a conic all meet in a point.
90. Brianchon's theorem may also be used to solve the following problems:
Given four tangents and the point of contact on any one of them, to construct other tangents to a conic. Given three tangents and the points of contact of any two of them, to construct other tangents to a conic.