the tangent line at that point. Pascal’s theorem thus affords a ready
method of drawing the tangent line to a conic at a given point. If the
conic is determined by the points 1, 2, 3, 4, 5 (Fig. 15), and it is
desired to draw the tangent at the point 1, we may call that point 1, 6.
The points _L_ and _M_ are obtained as usual, and the intersection of 34
with _LM_ gives _N_. Join _N_ to the point 1 for the desired tangent at
that point.




*76. Inscribed quadrangle.* Two pairs of vertices may coalesce, giving an
inscribed quadrangle. Pascal’s theorem gives for this case the very
important theorem

_Two pairs of opposite sides of any quadrangle inscribed in a conic meet
on a straight line, upon which line also intersect the two pairs of
tangents at the opposite vertices._

[Figure 16]

FIG. 16


[Figure 17]

FIG. 17


For let the vertices be _A_, _B_, _C_, and _D_, and call the vertex _A_