If one takes a point with coordinates (b, c) in a plane and draws
tangents to the discriminant parabola drawn in that plane, the slopes of
the tangents are the roots of the equation x^2+bx+c=0. Another way of
saying it, the slope of every tangent line to the discriminant parabola
is a root of every possible real monic quadratic equation whose
coefficients b and c are the coordinates (b, c) of some point on the
line (outside of the discriminant parabola).
The three propositions are clearly related. So if you track down the
sources of one, it may well contain the sources for the other.
One caveat: the tangent property in coefficient space seems to be
generalizable to any dimension (and corresponding polynomials) have not
considered yet if the proposition posted by Clark Kimberling can be
generalized (it may be trivial, I simply have not looked at it).
Credit: The originator (TTBOMK) of the tangential root search is Gabriel
Katz.
VS-)
Clark Kimberling wrote:
>
> I'm passing this inquiry along:
>
> If you pick points (r,u) and (s,v) on a parabola, draw tangents, and
> see where they intersect, the x-coordinate is (r + s)/2. This property
> was recently observed to _characterize_ parabolas, but it is felt that
> the whole business may be "well known". Has anyone seen it before?