R. Honsberger,
Mathematical Gems, III,
MAA, 1985, pp. 24-26
An Old Japanese Theorem
In Roger Johnson's marvellous old geometry text-Advanced Euclidean Geometry, first published in 1929 - he reports (on page 193 of the Dover edition, 1960) on the ancient custom by Japanese mathematicians of inscribing their discoveries on tablets which were hung in the temples to the glory of the gods and the honor of the authors. The following gem is known to have been exhibited in this way in the year 1800.
Let a convex polygon, which is inscribed in a circle, be triangulated by drawing all the diagonals from one of the vertices, and let the inscribed circle be drawn in each of the triangles. Then the sum of the radii of all these circles is a constant which is independent of which vertex is used to form the triangulation (Figure 1).
A great deal more might have been claimed, for this same sum results for every way of triangulating the polygon! (Figure 2). As we shall see, a simple application of a beautiful theorem of L. N. M. Carnot (1753-1823) settles the whole affair.
First of all, observe that any triangulation of an n-gon with its diagonals consists of
r_{i} + R = OO_{i}
and the sum in question is given by
(*) | r_{1} + r_{2} + ... + r_{n-2} = OO_{1} + OO_{2} + ... + OO_{n-2} - (n - 2)R |
Note now that the sum OO_{1} + OO_{2} + ... + OO_{n-2} on the right consists of perpendiculars to the sides of the polygon each counted once, and perpendiculars to the diagonals each counted twice since every diagonal serves as a side of two adjacent triangles. Furthermore, considering such internal perpendiculars, each is taken once with the sign plus and another time with the sign minus thus contributing nothing to the sum. Therefore, this sum is actually equal to the sum of all perpendiculars from O to the sides of the polygon. And, therefore, the right-hand side in (*) is constant. So is the left-hand side.
Remark 1
There is another theorem that is also known as Japanese.
Remark 2
David Richeson of Dickinson College has extended (2011) the theorem to non-convex polygons.
References
- R. Honsberger, Mathematical Gems III, MAA, 1985
- R. A. Johnson, Advanced Euclidean Geometry (Modern Geometry), Dover, 1960
Sangaku
- Sangaku: Reflections on the Phenomenon
- Critique of My View and a Response
- 1 + 27 = 12 + 16 Sangaku
- 3-4-5 Triangle by a Kid
- 7 = 2 + 5 Sangaku
- A 49^{th} Degree Challenge
- A Geometric Mean Sangaku
- A Hard but Important Sangaku
- A Restored Sangaku Problem
- A Sangaku: Two Unrelated Circles
- A Sangaku by a Teen
- A Sangaku Follow-Up on an Archimedes' Lemma
- A Sangaku with an Egyptian Attachment
- A Sangaku with Many Circles and Some
- A Sushi Morsel
- An Old Japanese Theorem
- Archimedes Twins in the Edo Period
- Arithmetic Mean Sangaku
- Bottema Shatters Japan's Seclusion
- Chain of Circles on a Chord
- Circles and Semicircles in Rectangle
- Circles in a Circular Segment
- Circles Lined on the Legs of a Right Triangle
- Equal Incircles Theorem
- Equilateral Triangle, Straight Line and Tangent Circles
- Equilateral Triangles and Incircles in a Square
- Five Incircles in a Square
- Four Hinged Squares
- Four Incircles in Equilateral Triangle
- Gion Shrine Problem
- Harmonic Mean Sangaku
- Heron's Problem
- In the Wasan Spirit
- Incenters in Cyclic Quadrilateral
- Japanese Art and Mathematics
- Malfatti's Problem
- Maximal Properties of the Pythagorean Relation
- Neuberg Sangaku
- Out of Pentagon Sangaku
- Peacock Tail Sangaku
- Pentagon Proportions Sangaku
- Proportions in Square
- Pythagoras and Vecten Break Japan's Isolation
- Radius of a Circle by Paper Folding
- Review of Sacred Mathematics
- Sangaku à la V. Thebault
- Sangaku and The Egyptian Triangle
- Sangaku in a Square
- Sangaku Iterations, Is it Wasan?
- Sangaku with 8 Circles
- Sangaku with Angle between a Tangent and a Chord
- Sangaku with Quadratic Optimization
- Sangaku with Three Mixtilinear Circles
- Sangaku with Versines
- Sangakus with a Mixtilinear Circle
- Sequences of Touching Circles
- Square and Circle in a Gothic Cupola
- Steiner's Sangaku
- Tangent Circles and an Isosceles Triangle
- The Squinting Eyes Theorem
- Three Incircles In a Right Triangle
- Three Squares and Two Ellipses
- Three Tangent Circles Sangaku
- Triangles, Squares and Areas from Temple Geometry
- Two Arbelos, Two Chains
- Two Circles in an Angle
- Two Sangaku with Equal Incircles
- Another Sangaku in Square
- Sangaku via Peru
- FJG Capitan's Sangaku
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