How can two beams in the Michelson interferometer come back at the same time? I am trying to understand this concept from my textbook.
According to the figure below, the time difference between the two paths $\Delta t=0$.
But why is this the case? Looking at the picture, the distance travelled in path B-C is $$\sqrt{(ut)^{2}+L^{2}}$$ where $u$ is the speed of the moving plate and $t$ is time.
This distance should be longer than $L$ of path B-E.
So, if the speed of light is the same, shouldn't B-C path takes longer to travel?
 A: As @Steve notes in the comment to the OP, the full round-trip paths have to be considered.
With a Newtonian/Galilean analysis, 
the longitudinal round-trip B-E'-B' takes longer than the transverse round-trip B-C'-B'.
That's why they expected a time difference in the round-trips, which depends on the velocity. Rather than measure a tiny time-difference, this optical device used a phase-difference instead.
But that's not the result that was obtained in the experiment.
The observed time difference is zero.
(From on my comments at https://www.physicsforums.com/threads/michelson-and-morley-in-a-space-time-diagram-cant-make-it-work.1010417/ )
The signal along the longitudinal arm takes too long (compared to that of the transverse arm) to return to the moving source. The longitudinal arm is too far away for the receptions to coincide.
To have the reception events coincide (as suggested by the results of the experiment),
length contraction of the longitudinal arm is needed.
Here's a video of mine, using a spacetime diagram of the Michelson Morley apparatus,
suggesting that length contraction is needed
https://youtu.be/AXx3CB80rAk

Here's my interactive visualization
Relativity-LightClock-MichelsonMorley-2018 (robphy)
https://www.geogebra.org/m/XFXzXGTq
(The image shows the case without length contraction.)

Here is an old paper of mine, where I work out some of the geometry
[already treated in typical textbooks] on a spacetime diagram
"Visualizing proper-time in Special Relativity"
https://arxiv.org/abs/physics/0505134
