The three lines joining the three pairs of opposite vertices of a hexagon circumscribed about a conic meet in a point.

86.  Construction of the pencil by Brianchon’s theorem. Brianchon’s theorem furnishes a ready method of determining a sixth line of the pencil of rays of the second order when five are given.  Thus, select a point in line 1 and suppose that line 6 is to pass through it.  Then l = (12, 45), n = (34, 61), and the line m = (23, 56) must pass through (l, n).  Then (23, ln) meets 5 in a point of the required sixth line.

[Figure 22]

FIG. 22

87.  Point of contact of a tangent to a conic. If the line 2 approach as a limiting position the line 1, then the intersection (1, 2) approaches as a limiting position the point of contact of 1 with the conic.  This suggests an easy way to construct the point of contact of any tangent with the conic.  Thus (Fig. 22), given the lines 1, 2, 3, 4, 5 to construct the point of contact of 1=6.  Draw l = (12,45), m =(23,56); then (34, lm) meets 1 in the required point of contact T.

[Figure 23]

FIG. 23

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.

[Figure 24]

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.

91.  Harmonic tangents. We have seen that a variable tangent cuts out on any two fixed tangents projective point-rows.  It follows that if four tangents cut a fifth in four harmonic points, they must cut every tangent in four harmonic points.  It is possible, therefore, to make the following definition: 

Four tangents to a conic are said to be harmonic when they meet every other tangent in four harmonic points.

92.  Projectivity and perspectivity. This definition suggests the possibility of defining a projective correspondence between the elements of a pencil of rays of the second order and the elements of any form heretofore discussed.  In particular, the points on a tangent are said to be perspectively related to the tangents of a conic when each point lies on the tangent which corresponds to it.  These notions are of importance in the higher developments of the subject.