A question from the "Einstein's Postulates" section from Modern Physics by Tipler:
Suppose that $A', B',$ and $C'$ are at rest in frame $S'$, which moves with respect to $S$ at speed $v$ in the $+x$ direction. Let $B'$ be located exactly midway between $A'$ and $C'$. At $t' = 0$, a light flash occurs at $B'$ and expands outward as a spherical wave.
The question then goes on to ask for the difference in the arrival of the wave fronts at $A'$ and $C'$ according to an observer in $S$.
Of course, if we assume that $A'$ and $C'$ are equidistant to $B'$ according to an observer in $S$, then we can easily find the time it takes for the light to reach both points. However, how can one show that $B'$ is still their midpoint? That is my main question, and it is important because if $B'$ is not their midpoint according to an observer in $S$, then it complicates trying to find the difference in time. Moreover, I would like this to be shown without using the Lorentz transformation because this question was posed before this concept was introduced.
Thank you for your time.