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This page uses true randomness to pick a random set of coordinates for a location on the planet's surface and show it on OpenStreetMap.
Note that OpenStreetMap uses the Mercator projection, which means that areas closer to the poles appear larger on the map than the actually are, and areas very close to the poles cannot be shown at all. The distortion factor tells you how many times bigger the area you're looking at appears compared to an area at the equator (where there is no distortion).
The coordinates are given in decimal format. If you prefer degrees, minutes and seconds, you can use the FCC's handy conversion tool.
Thanks to Randall M! Gee for code to make the random coordinates uniformly distributed on the surface of the planet!