Suppose a gravitational system consists of two large masses and all other masses are negligible. Then somewhere between the two large masses would be a point where their gravitational effects cancel out. Such a point is known as a Lagrangian point. It is the existence of such points that enable satellites to maintain their seemingly perpetual orbits - and hence the GPS and geocaching.
For this puzzle consider the static situation in which three large masses are placed at the three corners A, B, C of an equilateral triangle ABC, each side of which is one statute mile. If the masses are equal then one would expect there to be a Lagrangian point at O, the center of the triangle. There is. We could check that out recalling that gravitational force is proportional to the mass and inversely proportional to the square of the distance.
The surprising thing is that there are three additional Lagrangian points, say X, Y and Z.
Although just having the coordinates of A and O alone would be enough to determine a solution, those for the other two vertices of the triangle as well as for the Lagrangian points X and Y are given below. The cache is at the remaining Lagrangian point Z.
Evince coordinate checker replaced by Certitude on 1/22/19.
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