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# Tetrahedron

## Question

The following three coordinates are vertices of a regular tetrahedron circumscribed by the earth (assume spherical). The answer is at the fourth.

P1: S 17.6829061279° E 175.937938962°
P2: N 26.7978332041° E 072.5860786255°
P3: S 52.1980005058° W 015.0157003564°

Anyone know how to solve this?

## 2 answers to this question

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There is a straightforward procedure for this.

The trick is to recognize the following fact. A regular tetrahedron with one vertex at the North Pole will have its other three vertices at the same south latitude spaced 120 degrees apart in longitude. The next thing to note is that latitude and longitude translate easily into polar coordinates at constant radius. The final thing you have to understand is how to do rotations in spherical coordinates. These equations you can find in any standard text, or Googol them. Then you do following procedure.

Rotate your three given points until one of them, say Point 2 because it's closest, becomes the North Pole. The other two will now be at the same south latitude and separated by 120 degrees in longitude. Find the 3rd point 120 degrees from the other two at that latitude. That is the fourth point on the tetrahedron. Reverse the initial rotation, and you have your answer.

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P1: S 17.6829061279° E 175.937938962° magenta
P2: N 26.7978332041° E 072.5860786255° yellow
P3: S 52.1980005058° W 015.0157003564° cyan
P4: N 40.9 W082.38 white

with more or less 80km for 2 decimal places

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