Putnam 1987 A4

Let $P$ be a polynomial, with real coefficients, in three variables and $F$ be a function of two variables such that $$P(ux, uy, uz) = u^2 F(y-x,z-x) \text{for all real $x,y,z,u$},$$ and such that $P(1,0,0)=4$, $P(0,1,0)=5$, and $P(0,0,1)=6$. Also let $A,B,C$ be complex numbers with $P(A,B,C)=0$ and $|B-A|=10$. Find $|C-A|$.