394 12. MODERN GEOMETRIES12.4. Deduce Brianchon's theorem for a general conic from the special case of a
circle. How do you interpret the case of a regular hexagon inscribed in a circle?
12.5. Fill in the details of Plucker's proof of Brianchon's theorem, as follows: Sup-
pose that the equation of the conic is q(x, y) = y^2 + ð (x)y + r 2 (x) = 0, where n(x)
is a linear polynomial and r 2 (x) is quadratic. Choose coordinate axes not parallel
to any of the sides of the inscribed hexagon and such that the x-coordinates of all of
its vertices will be different, and also choose the seventh point to have x-coordinate
different from those of the six vertices. Then suppose that the polynomial gener-
ated by the three lines is s(x, y) = y^3 + h(x)y^2 + t 2 (x)y + t 3 (x) = 0, where tj(x) is
of degree j, j = 1,2,3. Then there are polynomials Uj(x) of degree j, j = 1,2,3,
such that
s(x, y) = q(x, y) (y - ui(x)) + (u 2 (x)y + u 3 (xj).
We need to show that u 2 = Q and u 3 = 0. At the seven points on the conic where
both q(x, y) and s(x, y) vanish it must also be true that u 2 (x)y+u 3 (x) — 0. Rewrite
the equation q(x, y) = 0 at these seven points as(u 2 y)^2 + riu 2 (u 2 y) + u\r 2 = 0
observe that at these seven points u 2 y = —u 3 , so that the polynomial u 3 — ðu 2 u 3 +
u 2 r 2 , which is of degree 6, has seven distinct zeros. It must therefore vanish iden-
tically, and that means that
(2u 3 - riu 2 )^2 = u(r\ - 4r 2 ).
This means that either u 2 is identically zero, which implies that u 3 also vanishes
identically, or else u 2 divides u 3. Prove that in the second case the conic must be
a pair of lines, and give a separate argument in that case.
12.6. Consider the two equations
xy = 0,
x{y-\) = 0.Show that these two equations are independent, yet have infinitely many common
solutions. What kind of conic sections do these equations represent?
12.7. Consider the general cubic equation
Ax^3 + Bx^2 y + Cxy^2 + Dy^3 + Ex^2 + Fxy + Gy^2 + Hx + Iy + J = 0,which has 10 coefficients. Show that if this equation is to hold for the 10 points
(1,0), (2,0), (3,0), (4,0), (0,1), (0,2), (0,3), (1,1), (2,2), (1, -1), all 10 coefficients
A,..., J must be zero. In general, then, it is not possible to pass a curve of degree 3
through any 10 points in the plane. Use linear algebra to show that it is always
possible to pass a curve of degree 3 through any nine points, and that the curve is
generally unique.
On the other hand, two different curves of degree 3 generally intersect in 9
points, a result known as Bezout's theorem after Etienne Bezout (1730-1783), who
stated it around 1758, although Maclaurin had stated it earlier. How does it happen
that while nine points generally determine a unique cubic curve, yet two distinct
cubic curves generally intersect in nine points? [Hint: Suppose that a set of eight
points {(XJ,yj) : j = 1,...,8} is given for which the system of equations for
A,..., J has rank 8. Although the system of linear equations for the coefficients is
generally of rank 9 if another point is adjoined to this set, there generally is a point