Ladd, The Pascal Hexagram.
11
in this hexagon,
is the reciprocal of the conic
with respect to a third conic;
twelve points of which may be obtained by taking on each side of the Brianchon hexagon the two points which form a harmonic range with each of the two pairs of vertices on this side; for instance, on
the two points which are harmonic at once with
and with
The hexagon
is the reciprocal with respect to the conic
of the hexagon formed by joining its alternate vertices; the point
is the pole of the line
the point
is the pole of of the line
hence the Pascal
is the polar of the point
is the pole of
is the pole of
hence the Pascal
is the polar of the point
It follows that the intersection of the Pascals
which is the Kirkman
is the pole of a line joining
to
which is the Pascal
But the six hexagons,
form, by connectors of alternate vertices, a Brianchon hexagon composed of the same sides in different orders, and hence circumscribed to the same conic, therefore the six Pascals
are the poles of the six Kirkmans
with respect to the same conic
Moreover, the points
and
in which the first three and the second three Pascals intersect are the poles respectively of the lines
and
which connect the first three and the second three Kirkmans. The two
points in question are harmonic conjugates with respect to the conic
hence their polars with respect to
the
lines of the same notation, are harmonic conjugates with respect to the reciprocal conic,
The triangle whose vertices are two corresponding
points and the intersection of the
lines through them (or, what is the same thing, the triangle whose sides are two corresponding
lines and the line joining the
points on them) is a triangle self-conjugate with respect to the conic
two of its vertices being at the same time conjugate with respect to
and two of its sides with respect to
Since this conic,
is inscribed in the triangles
and
we shall call it the conic
(where the order of the letters in each group of three is of no consequence) and the conic with respect to which it is the reciprocal of
we shall call
There are ten conics
the