Mohr–Mascheroni theorem

Every straightedge-and-compass construction of points in the Euclidean plane can be carried out with a compass alone, assuming suitable initial points.

Compass-and-straightedge construction: A geometric construction using an unmarked straightedge and compass to locate points through lines and circles. The theorem asserts that every point constructible by this method is also compass-constructible.
Georg Mohr: A Danish mathematician who published a proof of compass-only geometric construction in 1672. His overlooked book gave the first known proof of the theorem.
Compass: A geometric instrument that draws circles centered at a chosen point with a chosen radius. It is the sole drawing instrument allowed by the theorem.
Straightedge-only construction: A geometric construction using an unmarked straightedge without a compass. Unlike compass-only construction, this method cannot generally recover arbitrary Euclidean constructions.
Circle-circle intersection: The points shared by two circles, found by comparing their centers and radii. Compass-only constructions obtain new points chiefly as intersections of circles.
Euclid's Elements: A foundational Greek mathematical treatise that organized geometry into definitions, propositions, and proofs. Classical Euclidean construction problems supplied the setting for the theorem.
Unmarked straightedge: A ruler-like instrument used to draw a line through two known points, without measuring distances. The theorem shows its construction power is redundant when only points are sought.
Poncelet–Steiner theorem: The theorem that a straightedge alone suffices for Euclidean constructions when one circle and its center are given. Its instrument-reduction result is complementary, but depends on an initial circle and center.
Perpendicular bisector: The line through a segment's midpoint that is perpendicular to the segment. Compass-only circle intersections can construct its defining midpoint and perpendicular direction.
Euclidean geometry: The geometry of points, lines, distances, and angles on a flat plane. The theorem concerns constructions within this geometric system.

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