Are the divided differences $f[a, b, c]$ and $f[a, c, b]$ equal to eqchother?

Question

I was wondering whether the divided differences $f[a, b, c]$ and $f[a, c, b]$ were equal. I tried proving this by writing out the definition and got that those two are equal iff $-cf(b) + cf(a) + af(b) = -bf(c) + bf(a) + af(c)$, but I don't know if what I did is correct. Now it is entirely possible that I did something stupid and that there is an obvious solution to this that I completely missed.

Answer 1

If $x_0,x_1,...,x_n$ are distinct points then the divided difference $f[x_0,x_1,...,x_n]$ is the leading coefficient in the interpolation polynomial of degree at most $n$ at the points $x_0,x_1,...,x_n$. This pretty much makes it obvious that the order of the points doesn't matter.