the point 1, 6; _B_, the point 2; _C_, the point 3, 4; and _D_, the point
5 (Fig. 16). Pascal’s theorem then indicates that _L = AB-CD_, _M =
AD-BC_, and _N_, which is the intersection of the tangents at _A_ and _C_,
are all on a straight line _u_. But if we were to call _A_ the point 2,
_B_ the point 6, 1, _C_ the point 5, and _D_ the point 4, 3, then the
intersection _P_ of the tangents at _B_ and _D_ are also on this same line
_u_. Thus _L_, _M_, _N_, and _P_ are four points on a straight line. The
consequences of this theorem are so numerous and important that we shall
devote a separate chapter to them.




*77. Inscribed triangle.* Finally, three of the vertices of the hexagon
may coalesce, giving a triangle inscribed in a conic. Pascal’s theorem
then reads as follows (Fig. 17) for this case:

_The three tangents at the vertices of a triangle inscribed in a conic
meet the opposite sides in three points on a straight line._

[Figure 18]

FIG. 18




*78. Degenerate conic.* If we apply Pascal’s theorem to a degenerate
conic made up of a pair of straight lines, we get the following theorem
(Fig. 18):