I understand this intuitively, but I'm having trouble with the proof. Say I have a plane:
$$Ax+By+Cz = D,$$ and I choose points on the plane:
$$(x_1, y_1, z_1), \space (x_2, y_2, z_2).$$
I know that the vector $(x_2 - x_1, y_2 - y_1, z_2 - z_1)$ must be perpendicular to the vector I end with, which I'll call $(a, b, c)$.
$$(a, b, c) \cdot (x_2 - x_1, y_2 - y_1, z_2 - z_1) = 0,$$ therefore $$(ax_2 - ax_1 + by_2 - by_1 + cz_2 - cz_1) = 0.$$ Thus, $$ax_2 + by_2 + cz_2 = ax_1 + by_1 + cz_1.$$ Since they're both on the plane, we know that $$Ax_2 + By_2 + Cz_2 = Ax_1 + By_1 + Cz_1 = D,$$ leaving me with:
$$Ax_2 + By_2 + Cz_2 = Ax_1 + By_1 + Cz_1$$
$$ax_2 + by_2 + cz_2 = ax_1 + by_1 + cz_1$$
but that isn't enough to justify that $A = a, B = b,$ and $C = c$, is it?