Pompeiu's theorem

For any point in the plane of an equilateral triangle, its distances to the three vertices satisfy the triangle inequalities and can serve as the side lengths of a triangle.

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Triangle inequality: In any metric space, the distance between two points is no greater than the sum of their distances to a third point. After rotating one vertex-to-point segment by 60°, this inequality bounds the third distance by the other two.
Dimitrie Pompeiu: A Romanian mathematician known for work in analysis and geometry. The theorem bears his name and is commonly attributed to him.
Triangle construction: The geometric process of building a triangle from specified measurements such as sides or angles. The theorem guarantees that the three distances provide valid side lengths for a construction.
Routh's theorem: A theorem giving the area of a triangle formed by three cevians in terms of their division ratios. It is a distinct named triangle theorem, unlike Pompeiu's distance-based claim.
Equilateral triangle: A triangle whose three sides have equal length and whose three angles measure 60 degrees. Its 60-degree rotational symmetry makes the distance comparison work.
Morley theorem: A theorem stating that the intersections of adjacent trisectors of a triangle form an equilateral triangle. It is another named result connecting triangle geometry with equilateral structure.
Distance geometry: The study of geometric configurations through the distances between their points. The theorem translates a planar point configuration into a condition on three distances.
Van Aubel's theorem: A theorem about the line segments joining centers of squares constructed on the sides of a quadrilateral. It is another geometry result named for Pompeiu's contemporary, but concerns a different configuration.
Rotation (geometry): A transformation that turns points around a fixed center through a specified angle. Rotating a segment through 60° aligns it with an edge of the equilateral triangle.
Euclidean geometry: The study of geometric figures and relations in a flat space governed by Euclid's postulates. The theorem is a classical result about distances and triangles in the plane.