A more detailed graphic and argument as to why k is constant. (If I ever get the latex to work :-( )Light travels in straight lines. So we can use geometry.
Trinagle (O,t,kt) and (O,2t, 2kt) are similar. Therefore we know that if light is emitted at the source at time t and arrives at the destination at time time kt, that light emitted at time 2t must arive at time 2kt - by geometry, and the fact that light travels in straight lines.
How do we know the triangles are similar? The included angle is obviously the same, and the two light rays (t,kt) and (2t,2kt) must be parallel. The light rays must be parallel because they must cover equal distances (x-coordinate distance) in equal times (t-coordinate times), meaning that their slope, dx/dt, must be the same.
The argument that light emitted from the destionationat time kt arrives back at the source at time k^2 t is based on the physical argument that the receiver and the transmitter can be interchanged, as there is no "absolute" motion, only relative motion.
We can use similar triangles again to show that
(O, kt, k^2t) and (O, 2kt, 2k^2t) are similar, completing the argument.
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