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602
PROBLEMS AND SOLUTIONS.
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parameters; but when two of these are properly determined, the points A, B, G and
their tangentials A', B', C' will be related as in the theorem; viz., the line through
any two of the points will pass through the tangential of the third point. The
analytical investigation is as follows :
Let the equations of the three tangents be x = 0, y = 0, 2 = 0, and suppose that,
for the points A, B, G respectively, we have
0 = 0, y = Xz), (y = 0, z = yx), 0=0, x = vy),
then the equation of a cubic touching the three lines at the three points respectively
will be
(y — Xzf (v 2 By + Gz) + (z — ixocf (X-Gz + Ax) + (x — vy) 2 (jAAx + By)
— \x 2 Aa? — v 2 By 3 — X 2 Cz 3 + Kxyz = 0,
where A, B, G, K are arbitrary coefficients; but if A : B : G = X : ¡i : v, then the
equation is
(y — Xzf V (fivy + Z) + (Z — fJ,x) 2 X (VXZ + X) + (CC — vy) 2 {XfJbX + y)
— X/Aa? — fxv-y 3 — vX 2 z 3 + Kxyz = 0,
where
A, A' are the intersections of x = 0, by y — Xz = 0, ^vy +z = 0 respectively,
B, B' „ „ y — 0, „ z — fix = 0, vXz + x = 0 „ ,
G, G' „ „ z =0, „ x-vy = 0, Xy,x + y = 0 „ ;
the equations of BG, GA, AB thus are
— fxx + y,vy + z = 0, x — vy + vXz = 0, Xpx + y — Xz = 0,
which pass through A', B', and C' respectively.
If we consider along with the points A, B, G the points J, K, L, and their
respective tangentials, then we have inscribed in the cubic a hexagon ALBJGK which
has the following properties, viz., the pairs of opposite sides and the three diagonals
pass through the three real inflexions in lined, viz.,
AL,
JG,
BK,
through y
LB,
GK,
JA,
„ a
BJ,
KA,
CL,
» d.
This shows that the six points A, B, G, J, K, L are the intersections of the cubic
by a conic; and moreover, considering the triangles ABG, JKL formed by the alternate
vertices, then in each triangle the sides pass through the tangentials of the opposite
vertices respectively.
In what precedes we have in effect found the coordinates (z, x, y) of the third
point of intersection with the cubic, of the line joining the points (y, z, x) and
