Details: The three vertices are:
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Vertex #1. lat/lon 44º27.609'N 093º30.175'W, UTM: 15T, 459991 E, 4923106 N.
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Vertex #2. lat/lon 44º26.661'N 093º19.582'W, UTM: 15T, 474030 E, 4921280 N.
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Vertex #3. lat/lon 44º23.215'N 093º19.641'W, UTM: 15T, 473926 E, 4914900 N.
Note that the use of the UTM coordinates may make some calculations easier! Vertex #1 is located north of the public access to Cody Lake, with Phelps Lake just across the road. Vertex #2 is near Union Lake and Mud Lake (each of which has an interesting cache. See the links below). Vertex #3 is south of the public access on the south side of Fox Lake. Although you need not go to all of these lakes to obtain the cache each is scenic, interesting and worth a visit.
The Incenter. Three points, if they're not in a straight line, determine a triangle. A circle, called the incircle, can be inscribed inside the triangle in such a way that it osculates (barely touches and is tangent to) each side. The center of that inscribed circle is known as the incenter of the triangle.
One way to determine the incenter then would be to solve a system of nine equations in nine unknowns. The unknowns are the two coordinates for each of the three points of tangency together with the coordinates of the center of the circle as well as its radius. The nine equations come from the requirements that the circle has to go through each of the three points of tangency; it must be tangent there, that is, the slopes must match; and each point of tangency must lie on a corresponding side of the triangle. Whew! When solving this system for our data, one will obtain, in addition to the actual solution, several extraneous solutions arising from the possibility that the lines comprising the sides of the triangle can go on forever. The desired solution is the one closest to the parking lot indicated above.
Another, fortunately simpler, way to calculate the incenter is to use the fact that the incenter is the intersection of the angle bisectors of the triangle. Intersecting any two of the lines that bisect the angles of the triangle gives a simple system of two equations in two unknowns, and yields a unique solution which is the desired incenter.
There may be other ways to obtain the incenter. If you're successful in locating the cache, please include your method in your log. Thanks.
(take a peek at the circular house located at 4601 Circle Lake Tr.)
You can check your answers for this puzzle on Geochecker.com.
Other Links:
Other Triangle Center caches are at:
Other caches featuring the lakes of western Rice county are (from north to south):
Winter Friendliness: Should be able to handle up to a foot or so of snow.