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Here is my problem that I'm having some trouble with: I have the coordinates (latitude and longitude) of two points on Earth. I have no problem finding the great circle distance between the two locations using the spherical law of cosines. What I would like to do, however, is find a third location in terms of latitude and longitude that is the same distance from the two known locations.

My initial thought was to set the equation for the distance between the first known location and the unknown location equal to the equation for the distance between the second known location and the unknown location, but then things get kind of messy for me when trying to solve for the unknown latitude and longitude. Is this the right approach? Any help with this would be appreciated!

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    $\begingroup$ There isn't one single point that's equidistant from both points. If you find the great circle on which your two initial points fall, then the points which are equidistant from your first two are going to fall on a great circle orthogonal to your first one. Can you figure out how to construct that? $\endgroup$
    – Ian Coley
    Mar 29, 2013 at 17:05
  • $\begingroup$ Oops I wasn't thinking right. Instead, let's say I know the distance I want the unknown point to be from each of the two points, which should give me just two points as answers. How could I find the latitude and longitude of those? $\endgroup$
    – user70048
    Mar 29, 2013 at 17:13
  • $\begingroup$ Gotcha. I'm not too great on my latitude and longitude, but I imagine once you find the great circle on which your points must lie, you have a formula for the latitude and longitude of all those points. Does that help? $\endgroup$
    – Ian Coley
    Mar 29, 2013 at 17:42
  • $\begingroup$ This question still confuses me. Are you saying the distance of the third point from the other two points is a parameter to problem, or that you want the third point's distance from either point to be equal to the distance between the original two points (ie, the equilateral solution suggested below)? $\endgroup$
    – user2469
    May 13, 2013 at 3:33

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This may just be my lack of classical education speaking, but to me the sane and simple thing would be first to convert your spherical coordinates to 3D rectangular coordinates (for two points on the unit sphere), then solve the problem in 3D, and finally convert the resulting point back to spherical coordinates.

In rectangular coordinates, the points you're looking for is at the intersection between three spheres, namely the unit sphere where everything takes place, and the spheres centered on each of your two points containing the other one.

Write down the equations for these three spheres, and subtract two of them. This makes the $x^2$, $y^2$, $z^2$ terms cancel out, and you're left with the equation for a plane that contains your target point. Do the same thing for a different pair of spheres, giving you a different plane. The target point must lie on the line where those two planes intersect; finding a parametric equation for that line is routine. Then you can find the intersection of that line with one of the spheres, which is just a quadratic equation in the parameter. In the general case there will be two solutions, corresponding to putting your equilateral spherical to the left or to the right of the line segment you start out with.

Note that there may be no solution at all; for example two antipodal points are not the corners of any spherical equilateral triangle.

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