If we have sampled a Brownian motion at $t_i$, how can we get samples at the midpoints of $[t_{i-1},t_i]$ using a Brownian bridge? Suppose we have sampled a Brownian motion $(B_t)_{t\ge0}$ at $0=t_0<t_1<\cdots$. How can we obtain a sample at the midpoints of $[t_{i-1},t_i]$ from those samples? I've read that this is possible by a Brownian bridge, but wasn't able to figure out how exactly.

The triplet $(B(t_1),B(t_2),B(t_3))$ is jointly distributed as a Normal vector
$$\mathcal N_3(0_3,\Sigma)\quad\text{with}\quad\Sigma=\left[\begin{matrix} 
t_1 &t_1 &t_1\\
t_1 &t_2 &t_2\\
t_1 &t_2 &t_3
\end{matrix}\right]
\quad\text{when}\ \ \ t_1<t_2<t_3$$
Hence the conditional distribution of $B(t_2)$ conditional on $(B(t_1),B(t_3))$ [which are assumed know by the very definition of a Brownian bridge] is given by
$$B(t_2)\mid(B(t_1),B(t_3))=(x,y)\sim\mathcal N_1\left(\frac{(t_3 − t_2)x + (t_2-t_1)y}{t_3 − t_1} ,\ \frac{(t_2 − t_1)(t_3 − t_2)}{t_3 − t_1}\right).$$
