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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 equal incircles theorem derives from a Japanese
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 ...
, and pertains to the following construction: a series of rays are drawn from a given point to a given line such that the inscribed circles of the triangles formed by adjacent rays and the base line are equal. In the illustration the equal blue circles define the spacing between the rays, as described. The theorem states that the incircles of the triangles formed (starting from any given ray) by every other ray, every third ray, etc. and the base line are also equal. The case of every other ray is illustrated above by the green circles, which are all equal. From the fact that the theorem does not depend on the angle of the initial ray, it can be seen that the theorem properly belongs to
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
, rather than geometry, and must relate to a continuous scaling function which defines the spacing of the rays. In fact, this function is the
hyperbolic sine In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the un ...
. The theorem is a direct corollary of the following lemma: Suppose that the ''n''th ray makes an angle \gamma_n with the normal to the baseline. If \gamma_n is parameterized according to the equation, \tan \gamma_n = \sinh\theta_n, then values of \theta_n = a + nb, where a and b are real constants, define a sequence of rays that satisfy the condition of equal incircles, and furthermore any sequence of rays satisfying the condition can be produced by suitable choice of the constants a and b.


Proof of the lemma

In the diagram, lines PS and PT are adjacent rays making angles \gamma_n and \gamma_ with line PR, which is perpendicular to the baseline, RST. Line QXOY is parallel to the baseline and passes through O, the center of the incircle of \triangle PST, which is tangent to the rays at W and Z. Also, line PQ has length h-r, and line QR has length r, the radius of the incircle. Then \triangle OWX is similar to \triangle PQX and \triangle OZY is similar to \triangle PQY, and from XY = XO + OY we get : (h-r) ( \tan \gamma_ - \tan \gamma_n ) = r ( \sec \gamma_n + \sec \gamma_ ). This relation on a set of angles, \, expresses the condition of equal incircles. To prove the lemma, we set \tan \gamma_n = \sinh (a+nb), which gives \sec \gamma_n = \cosh(a+nb). Using a+(n+1)b = (a+nb)+b, we apply the addition rules for \sinh and \cosh, and verify that the equal incircles relation is satisfied by setting : \frac = \tanh\frac. This gives an expression for the parameter b in terms of the geometric measures, h and r. With this definition of b we then obtain an expression for the radii, r_N, of the incircles formed by taking every ''N''th ray as the sides of the triangles : \frac = \tanh\frac{2}.


See also

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Hyperbolic function In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
*
Japanese theorem for cyclic polygons __notoc__ In geometry, the Japanese theorem states that no matter how one triangulates a cyclic polygon, the sum of inradii of triangles is constant.Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929). Conversel ...
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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 ...
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Tangent lines to circles In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...


References


Equal Incircles Theorem
at cut-the-knot *J. Tabov. A note on the five-circle theorem. ''Mathematics Magazine'' 63 (1989), 2, 92–94. Euclidean geometry Japanese mathematics Theorems about triangles and circles