Constant chord theorem

In a circle, chords subtending the same inscribed angle have equal lengths. The result follows because equal inscribed angles intercept equal arcs and therefore equal chords.

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Inscribed angle theorem: An inscribed angle measures half the measure of its intercepted arc. Equal inscribed angles intercept equal arcs, the key step connecting angle size to chord length.
Circle: The set of points in a plane at a fixed distance from a center. The theorem concerns chords within one circle, where a common radius makes arc and chord comparisons possible.
Cyclic quadrilateral: A quadrilateral whose four vertices lie on one circle. Its equal angles can imply equal opposite-side chords through the theorem.
Perpendicular bisector theorem: A point lies on a segment's perpendicular bisector exactly when it is equidistant from the segment's endpoints. It proves equal chords through distances from the center, rather than through inscribed angles.
Congruent chords: Chords of the same circle are congruent exactly when they intercept equal minor arcs. The theorem turns equality of intercepted arcs into equality of chord lengths.
Chord: A line segment whose endpoints lie on a circle. A chord is the geometric object whose length the theorem equates.
Rectangle: A quadrilateral with four right angles. A rectangle inscribed in a circle has equal diagonals because they subtend equal right angles.
Thales' theorem: An angle inscribed in a semicircle is a right angle. It fixes the angle at a special value, while the constant chord theorem compares equal arbitrary inscribed angles.
Central angle: An angle whose vertex is the center of a circle and whose sides are radii. A central angle measures its intercepted arc directly, clarifying the arc comparison behind the theorem.
Arc: A connected portion of a circle's circumference between two points. The equal arcs intercepted by equal inscribed angles provide the bridge to equal chords.