JesseM,
By Pythagoras' theorem ...
It's just from Einstein's kinematic model ... in a 2-space diagram of system K (ie. x,y,z) with time (t) implied, imagine the system
k (ie X,Y,Z, time Tau implied) moving at inertial v along +x with x & X colinear and y & Y always parallel. The origin is marked by the momentary colocation of both system origins at t=Tau=0. Imagine a spherical EM pulse emitted from said origin, and a subsequent consideration of the intersection of said expanding EM sphere with the moving Y-axis in quadrant 1 (ie +x,+y). This interval represents the path of a single photon from origin to the moving Y-axis at some arbitrary time t. This lightpath is defined in system K by Pathagorus' theorem ...
(ct)2 = (vt)2+y2
where x=vt is the location of the system
k origin in system K at time t. Since no length contractions exist wrt axes orthogonal to the direction of motion, then y=Y. Add, per system
k, Y=cTau, so y=Y=cTau. So ...
(ct)2 = (vt)2+y2
(ct)2 = (vt)2+(cTau)2
then solving for c, this becomes ...
c= vt/(t2-tau2)1/2