Haruki's theorem

Haruki's theorem is a Euclidean geometry result about segment lengths in configurations involving two intersecting circles. Its name is used for a circle-geometry relation associated with Japanese geometry problems.

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Intersecting circles: Two circles that share two distinct points. The theorem concerns length relations in configurations built from this basic arrangement.
Secant-secant theorem: A circle theorem equating products of segment lengths along two secants from one external point. Secant products are a standard route to length identities in circle configurations.
Two intersecting circles: A configuration of two circles crossing at two points, with their centers and common chord specified as needed. This is the theorem's defining geometric setting.
Japanese geometry: The study and teaching of geometry in Japan, including its historical traditions and modern practice. The theorem's name suggests a connection to Japanese mathematical naming or exposition.
Chord: A line segment whose endpoints lie on a circle. Chords provide the circle segments whose lengths enter the theorem's configurations.
Ptolemy's theorem: A relation among the four sides and two diagonals of a cyclic quadrilateral. Cyclic quadrilaterals can turn circle intersections into tractable segment equations.
Common chord: A chord shared by two circles that intersect at two points. The shared chord is the most direct segment structure in the theorem's circle arrangement.
Sangaku: Japanese wooden votive tablets bearing geometric problems, offered at Shinto shrines and Buddhist temples. Historical Japanese geometry provides context for named circle-geometry problems.
Power of a point: A signed quantity relating a point's distance from a circle's center to its radius. It organizes products of lengths formed by secants through a point.
Similar triangles: Triangles with equal corresponding angles and proportional corresponding side lengths. Angle-chasing often reduces circle geometry length claims to proportional sides.