We study two processes obtained as follows: Take two independent $d$-dimensional Brownian motions started at points $x, y$, respectively. For the first process, let $d \geq 3$ and condition on $X_t = Y_t$ for some $t$ (a set of probability 0). Run $X$ out to the point of intersection and then run $Y$ in reversed time from this point back to $y$. For the second process, let $d \geq 5$ and perform the same construction, this time conditioning on $X_s = Y_t$ for some $s, t$. The first process is shown to be Doob's conditioned (to go from $x$ to $y$) Brownian motion $Z$, and the second has distribution absolutely continuous with respect to that of $Z$, the Radon-Nikodym density being a constant times the time $Z$ takes to travel from $x$ to $y$. Similar results (including extensions to the critical dimensions $d = 2$ and $d = 4$) are obtained by conditioning the motions to hit before they leave domains. We use the asymptotics of the probability of "near misses" and results on the weak convergence of $h$-transforms.