The velocity addition problem, solved

Recall the Pythagorean Rule (which the Mesopotamians knew at least a millenium before the Pythagoreans). By taking differentials the Pythagorean Rule can be written as what is called the Euclidean metric
dL2 = dX2 + dY2
(Note: this is for two space dimensions. In higher space dimensions you just add more terms to the formula. In three space dimensions it's dL2 = dX2 + dY2 + dZ2.)
Now what to do about spacetime?
It turns out that finally about 4000 years after the Mesopotamians experimentally (but not theoretically) figured out the Euclidean metric, Albert Einstein hit on the spacetime analog, which we now call the Minkowski metric: Lorentz transformation
dS2 = c2 dT2 - dL2
where dL2 is the Euclidean metric on the space part of the spacetime. For example, in two space and one time dimensions, the Minkowski metric becomes:
dS2 = c2 dT2 - dX2 - dY2
In the cases we've been looking at, the relevant Minkowski metric is simply
dS2 = c2 dT2 - dX2
Recall the Lorentz transformation (using b = (U/c) for shorthand):
c T = c T' / (1 - b2)1/2 + X' b/ (1 - b2)1/2

X = c T' b/ (1 - b2)1/2 + X' / (1 - b2)1/2
The Lorentz transformation has the amazing property that it leaves the Minkowski metric unchanged so that
c2 dT2 - dX2 = c2 dT' 2 - dX' 2,
which is demonstrated in the above figure. The two purple lines represent different values of the Lorentz-invariant interval dS2.
In particular, the bright purple line representing dS2 = 0 is important because it represents the path of light (which is described by both X = cT and X' = cT'). This shows that the Minkowski metric invariance under Lorentz transformations really does encode the observed constancy of the speed of light in Nature. So we're on the right track here in trying to model the natural world using the language of mathematics.
How does the Minkowski metric help us solve the velocity addition problem and come up with a new rule for adding velocities that will encode the observed behavior of the constant speed of light as a maximum observable speed in nature?
dX = dX' / (1 - b2)1/2 + c dT' b / (1 - b2)1/2 = dT' (V' + U)/(1 - b2)1/2

c dT = c dT' / (1 - b2)1/2 + dX' b /(1 - b2)1/2 = c dT' (1 + U V'/c2)/(1 - b2)1/2
The velocity we're looking for is V = dX/dT. Dividng the top line dX above by the bottom c dT, we get for V:
Relativistic velocity addition rule
dX/dT = V = (U + V')/(1 + U V'/c2)
as the velocity addition rule consistent with the observed behavior of nature.
Let's check: if the blue car is rolling at U = c/2 and a laser on the blue car is travelling at V' = c then the red driver sees the laser travelling at V = (c/2 + c)/(1 + 1/2) = c so this Minkowski metric models very well the observed behavior of Nature that one can't seem to make light travel faster than c, not by putting a laser on a moving car, or a rocket or any other type of moving vehicle. If either U=c or V'=c, the velocity addition formula reduces to V=c.
Isn't that clever? How could nature be that smart?