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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 no matter how one
triangulates a
cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in s ...
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two to ...
, the
sum of
inradii of
triangle
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- colline ...
s is
constant.
[Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929).]
Conversely, if the sum of inradii is independent of the triangulation, then the polygon is cyclic. The Japanese theorem follows from
Carnot's theorem; it is a
Sangaku problem.
Proof
This theorem can be proven by first proving a special case: no matter how one triangulates a cyclic ''quadrilateral'', the sum of inradii of triangles is constant.
After proving the quadrilateral case, the general case of the cyclic polygon theorem is an immediate corollary. The quadrilateral rule can be applied to quadrilateral components of a general partition of a cyclic polygon, and repeated application of the rule, which "flips" one diagonal, will generate all the possible partitions from any given partition, with each "flip" preserving the sum of the inradii.
The quadrilateral case follows from a simple extension of the
Japanese theorem for cyclic quadrilaterals
In geometry, the Japanese theorem states that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle.
Triangulating an arbitrary cyclic quadrilateral by its diagonals yields four overlapping t ...
, which shows that a rectangle is formed by the two pairs of incenters corresponding to the two possible triangulations of the quadrilateral. The steps of this theorem require nothing beyond basic constructive Euclidean geometry.
With the additional construction of a parallelogram having sides parallel to the diagonals, and tangent to the corners of the rectangle of incenters, the quadrilateral case of the cyclic polygon theorem can be proved in a few steps. The equality of the sums of the radii of the two pairs is equivalent to the condition that the constructed parallelogram be a rhombus, and this is easily shown in the construction.
Another proof of the quadrilateral case is available due to Wilfred Reyes (2002).
In the proof, both the
Japanese theorem for cyclic quadrilaterals
In geometry, the Japanese theorem states that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle.
Triangulating an arbitrary cyclic quadrilateral by its diagonals yields four overlapping t ...
and the quadrilateral case of the cyclic polygon theorem are proven as a consequence of
Thébault's problem III.
See also
*
Carnot's theorem, which is used in a proof of the theorem above
*
Equal incircles theorem
*
Tangent lines to circles
Notes
References
*Claudi Alsina, Roger B. Nelsen: ''
Icons of Mathematics: An Exploration of Twenty Key Images''. MAA, 2011, {{ISBN, 9780883853528, pp
121-125*Wilfred Reyes
''An Application of Thebault’s Theorem'' Forum Geometricorum, Volume 2, 2002, pp. 183–185
External links
*Mangho Ahuja, Wataru Uegaki, Kayo Matsushita
In Search of the Japanese Theorem
at Mathworld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Di ...
Japanese Theorem
interactive demonstration at the C.a.R. website
*Wataru Uegaki: "Japanese Theoremの起源と歴史" (On the Origin and History of the Japanese Theorem) http://hdl.handle.net/10076/4917
Euclidean plane geometry
Japanese mathematics
Theorems about triangles and circles