Van Schooten's theorem
For an equilateral triangle ABC and a point P on the minor arc BC of its circumcircle, the theorem states that PA = PB + PC.
Equilateral triangle: A triangle whose three sides have equal length and whose three angles each measure 60 degrees. Its 120-degree rotational symmetry underlies the segment construction used in the proof.
Minor arc: The shorter of the two arcs between distinct points on a circle. The theorem specifies this arc so P occupies the relevant side of triangle ABC.
Euclidean geometry: The geometry of points, lines, angles, and distances in a plane or three-dimensional space. Van Schooten’s result is a metric theorem within the Euclidean plane.
Circumcenter: The center of a triangle’s circumcircle, equidistant from all three vertices. For an equilateral triangle, choosing P at this center makes the distance identity immediate.
Circumcircle: The unique circle passing through all three vertices of a nondegenerate triangle. The position of P on this circle fixes the equal-angle relations in the theorem.
Central angle: An angle whose vertex is the center of a circle and whose sides are radii. The central angles subtending the sides of an equilateral triangle are each 120 degrees.
Geometric inequality: An inequality relating geometric quantities such as lengths, areas, or angles. The identity can be used to establish bounds when the three distances appear in a larger problem.
Arc midpoint: The point halfway along a specified arc of a circle. At the midpoint of minor arc BC, PB and PC are equal, giving a symmetric instance.
Rotation (geometry): A transformation that turns every point in a plane through the same angle about a fixed center. A 60-degree rotation can map a vertex and segment to a configuration yielding the length sum.
Chord (geometry): A line segment joining two points on a circle. PA, PB, and PC are chords, so their lengths encode the arcs between their endpoints.