Japanese Theorem For Cyclic Quadrilaterals
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In
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the Japanese theorem states that the centers of the
incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter ...
s of certain
triangles A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensiona ...
inside a
cyclic quadrilateral In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertex (geometry), vertices all lie on a single circle, making the sides Chord (geometry), chords of the circle. This circle is called ...
are vertices of a
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...
. It was originally stated on a
sangaku Sangaku or san gaku () are Japanese Euclidean geometry, geometrical problems or theorems on wooden tablets which were placed as offerings at Shinto shrines or Buddhist temples in Japan, Buddhist temples during the Edo period by members of all so ...
tablet on a temple in
Yamagata prefecture is a Prefectures of Japan, prefecture of Japan located in the Tōhoku region of Honshu. It has a population of 1,005,926 (1 February 2025) and an area of 9,325 Square kilometre, km2 (3,600 Square mile, sq mi). Its neighbours are Akita Prefectu ...
, Japan, in 1880. Triangulating an arbitrary cyclic quadrilateral by its diagonals yields four overlapping triangles (each diagonal creates two triangles). The centers of the incircles of those triangles form a rectangle. Specifically, let be an arbitrary cyclic quadrilateral and let , , , be the incenters of the triangles , , , . Then the quadrilateral formed by , , , is a rectangle. Proofs are given by Bogomolny and Reyes. This theorem may be extended to prove the Japanese theorem for cyclic polygons, according to which the sum of inradii of a triangulated cyclic polygon does not depend on how it is triangulated. The special case of the theorem for quadrilaterals states that the two pairs of opposite incircles of the theorem above have equal sums of radii. To prove the quadrilateral case, simply construct the parallelogram tangent to the corners of the constructed rectangle, with sides parallel to the diagonals of the quadrilateral. The construction shows that the parallelogram is a rhombus, which is equivalent to showing that the sums of the radii of the incircles tangent to each diagonal are equal. This related result comes from an earlier sangaku tablet, also from Yamagata, from 1800. The quadrilateral case immediately proves the general case, as any two triangulations of an arbitrary cyclic polygon can be connected by a sequence of
flips Flip, FLIP, or flips may refer to: People * Flip (nickname), a list of people * Lil' Flip (born 1981), American rapper * Flip Simmons, Australian actor and musician * Flip Wilson, American comedian Arts and entertainment Fictional characters * ...
that change one diagonal to another, replacing two incircles in a quadrilateral by the other two incircles with equal sum of radii.


See also

* Carnot's theorem *
Japanese mathematics denotes a distinct kind of mathematics which was developed in Japan during the Edo period (1603–1867). The term ''wasan'', from ''wa'' ("Japanese") and ''san'' ("calculation"), was coined in the 1870s and employed to distinguish native Japanese ...


References


Further reading

*Mangho Ahuja, Wataru Uegaki, Kayo Matsushita: "In Search of the Japanese Theorem". In: ''Missouri Journal of Mathematical Sciences'', vol 18, no. 2, May 2006
online
at project Euclid) *Wataru Uegaki
""
(On the Origin and History of the Japanese Theorem). Departmental Bulletin Paper, Mie University Scholarly E-Collections, 2001-03-01 *{{Cite web, author=笹部貞市郎, url=https://archive.org/details/20240618_20240618_1431/page/119/mode/1up , title=几何学辞典: 问题解法 , website=
Archive.org The Internet Archive is an American non-profit organization founded in 1996 by Brewster Kahle that runs a digital library website, archive.org. It provides free access to collections of digitized media including websites, software applic ...
, date=1976 , at=Problem 587


External links


Japanese theorem, interactive proof with animation
Euclidean plane geometry Japanese mathematics Theorems about quadrilaterals and circles