Expectation of a brownian motion given two states

Question

I have a following question about standard brownian motion. For $0 < s < t < u$. How could I derive $E(B_t|B_s,B_u)$? Here is my try: First, $$E(B_t|B_s,B_u) = E(B_t\cdot B_s|B_u) / P(B_s|B_u)$$ For the numerator, we can calculate that it equals to $$s/t(t/u (u-t) - (t/uB_u)^2$$ Also, as $(B_s|B_u)\sim\mathcal{N}(s/u \cdot B_u, s/u(u-s))$, it is easy to derive $P(B_s|B_u)$. Is my approach correct? Do you guys have a better approach?