Napoleon's problem

The challenge is to divide the circle into four equal arcs using only a compass.

[1][2] Napoleon was known to be an amateur mathematician, but it is not known if he either created or solved the problem.

Napoleon's friend the Italian mathematician Lorenzo Mascheroni introduced the limitation of using only a compass (no straight edge) into geometric constructions.

But actually, the challenge above is easier than the real Napoleon's problem, consisting in finding the center of a given circle with compass alone.

The following sections will describe solutions to three problems and proofs that they work.

Note that the line segments OV, OX, OY, OZ, VX, XY, YZ have the same length, all distances being equal to the radius of the circle C. Now draw an arc centred on V which goes through Y and an arc centred on Z which goes through X; call where these two arcs intersect T. Note that the distances VY and XZ are

Point C can be determined from B and B', using circles of radius b. Triangle ABA' has a right angle at B and BH is perpendicular to AA', so : Therefore,