The Conway Circle
Take a triangle. Extend each side by the lengths of the other two sides, as in the figure where all segments of the same color are equal. Then the endpoints of the extended segments (with length the triangle's perimeter) lie on a circle centered at the incenter.
Johnson Horton Conway is not only celebrated for high-ranked results. We also owe him for creating beautiful mathematics made accessible to interested amateurs. His circle is an example. For more information on the other interesting properties, just surf the web.