Humming Birds

Ten years ago (before PTC) I posted the problem below in the old MathCad forum.

Today I see that it is still posted, however, the then attached MathCad sheet with the answer is no longer posted.

So, I again post the problem.

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PROBLEM

Given 4 humingbirds each located at the vertice of a unit tetrahedron.

The birds begin to fly such that all birds fly at the same velocity and:

(1) bird #1 always flys directly towards bird #2

(2) bird #2 always flys directly towards bird #3

(3) bird #3 always flys directly towards bird #4

(4) bird #4 always flys directly towards bird #1

As can be visualized, the flying birds ultimately meet and crash together. (Assume birds are very small.)

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Find: Length of the flight path for each bird