In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, 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 edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non-collinear ...
inside a
cyclic quadrilateral
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the ''circumcircle'' or ''circumscribed circle'', and the vertices are said to be ''c ...
are vertices of a
rectangle.
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.
Note that this theorem is easily extended to prove the
Japanese theorem for cyclic polygons. 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.
The quadrilateral case immediately proves the general case by induction on the set of triangulating partitions of a general polygon.
See also
*
Carnot's theorem
*
Sangaku
Sangaku or San Gaku ( ja, 算額, lit=calculation tablet) are Japanese geometrical problems or theorems on wooden tablets which were placed as offerings at Shinto shrines or Buddhist temples during the Edo period by members of all social classes ...
*
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
*Mangho Ahuja, Wataru Uegaki, Kayo Matsushita
''In Search of the Japanese Theorem''(postscript file)
Theoremat
Cut-the-Knot
*Wataru Uegaki
"{{nihongo2, Japanese Theoremの起源と歴史"(On the Origin and History of the Japanese Theorem). Departmental Bulletin Paper, Mie University Scholarly E-Collections, 2001-03-01
*Wilfred Reyes
''An Application of Thebault’s Theorem'' Forum Geometricorum, Volume 2, 2002, pp. 183–185
External links
Euclidean plane geometry
Japanese mathematics
Theorems about quadrilaterals and circles