Mascheroni's Construction of the Center of a Circle
How do you find the center of a circle with compass alone? Lorenzo Mascheroni found this beautiful and easy construction in 1797. He did not know that Dane G. Mohr had already discovered it in 1672. This Demonstration decomposes the steps of the construction. You start with two points. Justifying the result is not trivial, even with the help of the figure.
Lorenzo Mascheroni (1750–1800) asserted in his tract The Geometry of Compasses that ruler-and-compass constructions can be accomplished with the compass alone. Starting with two points, other points can be constructed. The solutions are uncomfortable, but they exist!
The construction is not only elegant but also quicker and more exact than using a compass and ruler.