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A
F
Z
E
B
D
C
Y
X
Figure 3.32.
Pascal's theorem.
L AF
L CD
B
E
C = D
Figure 3.33.
Points on a conic given
three points and two
tangents.
A = F
L
exist. The topic [HilC99] has a nice discussion about the connection between Pascal's
theorem and other results.
The converse of Pascal's theorem is actually more interesting because it allows
one to construct any number of points on a conic through five points no three of which
are collinear. Also, by letting the points A and F and the points C and D coalesce we
get a construction for the points on a conic given three points and the tangent lines
at the first two points. For example, in Figure 3.33 (which uses notation compatible
with that in Theorem 3.9.1) we are given the three points A = F , C = D , and B and
tangent lines L AF and L CD at the points A and C , respectively. The figure shows how
an arbitrary line L through the intersection of L AF and L CD determines a unique new
point E on the conic. Note that since the line L can be parameterized by the angle
that it makes with the line L AF , our construction also produces a parameterization of
the points on the conic.
The dual of Pascal's Theorem is called Brianchon's Theorem after C.J. Brianchon,
who proved it in 1806 before the principle of duality in P 2 was formulated. Of course,
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