Monge's theorem

Monge's theorem states that the three external centers of similitude formed by pairs of three circles or spheres lie on one straight line.

Center of similitude: A point from which a homothety maps one circle or sphere to another. Each pair of circles or spheres contributes an external center of similitude.
Circle: The set of points in a plane at a fixed distance from a center. The planar objects whose pairwise similitude centers the theorem aligns.
Gaspard Monge: A French mathematician and engineer who helped establish descriptive geometry. The theorem bears his name and reflects the geometry associated with his work.
Monge configuration: A configuration of three circles or spheres and their three pairwise external centers of similitude. It is the arrangement whose central collinearity is asserted by the theorem.
Homothety: A transformation that scales distances from a fixed center by a constant factor. The pairwise transformations underlying the centers in Monge's theorem.
Sphere: The set of points in three-dimensional space at a fixed distance from a center. The theorem extends from circles to spheres without changing its collinearity conclusion.
Descriptive geometry: A method of representing three-dimensional objects through geometric projections onto planes. Monge's broader geometric work helped shape the setting in which the theorem is taught.
Three-circle configuration: A planar arrangement of three circles whose pairwise external centers of similitude can be constructed. This is the two-dimensional case most often used to illustrate the result.
Menelaus's theorem: A theorem relating the ratios of directed segments formed when a transversal crosses a triangle's sides. Its directed-ratio relation proves collinearity of the three centers.
Collinearity: The property of points lying on one straight line. It is the geometric conclusion asserted for the three pairwise centers.

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