Given two points A and B, an ellipse is a set of all the points (C) such that the distance from Point C to Point A plus the distance from Point C to Point B is a fixed total. Points A and B are called the foci (pronounced FOE sigh - as in the Latin plural of focus). Two ellipses on the same plane can intersect at anywhere from 0 to 4 different points, as the above illustration shows.
To find the cache coordinates, you must identify the point(s) of intersection of two ellipses. These two ellipses have a common focus. The cache is located at a point such that it meets these two conditions:
The distance from Waypoint 1 to the cache plus the distance from Waypoint 2 to the cache is 5,000 meters.
The distance from Waypoint 2 to the cache plus the distance from Waypoint 3 to the cache is 10,000 meters.
Waypoint 1 is the location of my cache Circular Reasoning - the actual cache location, not the posted coordinates. Yes, that means you have to solve that puzzle before you can solve this one.
Waypoint 2: N37° 20.660 W122° 02.254
Waypoint 3: N37° 17.140 W122° 05.298
All distances were measured using Fizzymagic's FizzyCalc tool. This is a great little tool that all geocachers should have, even if you don't solve this puzzle. There is more than one solution to this puzzle. The correct solution does not require you to trespass on private property.