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 [pg 52] 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.
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.
Fig. 23