Cryptography Reference
In-Depth Information
2.4.1
The Theorems of Pappus and Pascal
Theorem 2.6 has two other nice applications outside the realm of elliptic
curves.
THEOREM 2.13 (Pascal's Theorem)
Let
ABCDEF
be a hexagon inscribed inacon csection (ellipse, parabola,
or hyperbola), w here
A, B,
C, D
, E,
F
ar
e distinct pointsinthe a n
e pl
ane.
Let
X
be the intersection of
AB
and
DE
,let
Y
be
the intersection of
BC
and
EF
,and let
Z
be the intersection of
CD
and
FA
.Then
X, Y, Z
are collinear
(see F igure 2.4).
Figure 2.4
Pascal's Theorem
REMARK 2.14
(1) A conic is given by an equation
q
(
x, y
)=
ax
2
+
bxy
+
cy
2
+
dx
+
ey
+
f
= 0 with at least one of
a, b, c
nonzero. Usually, it is assumed
that
b
2
−
4
ac
= 0; otherwise, the conic degenerates into a product of two linear
factors, and the graph is the union of two lines. The present theorem holds
even in this case, as long as the points
A, C, E
lie on one of the lines,
B, D, F
lie on the other
, an
d no
ne is
the intersection of the two lines.
(2) Possibly
AB
and
DE
are parallel, for example. Then
X
is an infinite
point in
P
2
K
.
(3) Note that
X, Y, Z
will always be distinct. This is easily seen as follows:
First observe that
X, Y, Z
cannot lie on the conic since a line can intersect
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