A bizarre but nice fact of polynomials

The tangent to the midpoint between two roots in a third degree polynomial(with three roots) is always the third root. Just saw that at Polymath.

Deriving f(x)=K*(x-a)*(x-b)*(x-c) using the product rule we get

f'(x)= K*((x-a)*(x-b)+(x-a)*(x-c)+(x-b)*(x-c))=

K*((x-a)*(x-b)+(x-c)*(2x-a-b))

taking x=v=(a+b)/2 the second term vanishes and we have

f'(v)= K*(v-a)*(v-b)

the crossing point should be

v - f(v)/f'(v)

as in Newton's method and in this case should be:

v - {K*(v-a)*(v-b)*(v-c)}/{K*(v-a)*(v-b)}

= v - (v - c) = c