*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