# Maximal Properties of the Pythagorean Relation

A 1893 sangaku from the Fukusima prefecture [Fukagawa, Pedoe, problem 2.3.6] admits a relatively straightforward computational solution, but also an elegant solution that practically avoids any kinds of computations.

Rectangles cut off from a right triangle leave three triangles with inradii r_{1}, r_{2}, r_{3}, in the increasing order. Show that when the area of the rectangle is a maximum possible then

(r_{1})² + (r_{2})² = (r_{3})².

### References

H. Fukagawa, D. Pedoe,

*Japanese Temple Geometry Problems*, The Charles Babbage Research Center, Winnipeg, 1989Write to:

Charles Babbage Research Center

P.O. Box 272, St. Norbert Postal Station

Winnipeg, MB

Canada R3V 1L6

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Copyright © 1996-2018 Alexander Bogomolny

Rectangles cut off from a right triangle leave three triangles with inradii r_{1}, r_{2}, r_{3}, in the increasing order. Show that when the area of the rectangle is a maximum possible then

(r_{1})² + (r_{2})² = (r_{3})².

### Solution

The four triangles ABC, EDC, AEF, DBG are obviously similar. Their inradii relate exactly like any of the corresponding sides. For the definiteness sake, let's focus on the hypotenuse.

If the sides of ΔABC are a = BC, b = AC, c = AB, then, for some

ABC | a | b | c | |||||

EDC | ka | kb | kc | |||||

AEF | ... | ... | (1-k)b | |||||

DBG | ... | ... | (1-k)a |

We want to show that, for k, for which the area of the rectangle DEFG is a maximum,

[(1 - k)a]² + [(1 - k)b]² = (kc)²,

or

(1) | (1 - k)²·a² + (1 - k)²·b² = k²·c². |

However, we also have

a² + b² = c².

Multiplying (2) by k² and subtracting from (1) we get

(1 - 2k)·(a² + b²) = 0

which is only possible when k = 1/2. This is then bound to be the optimal configuration implied by the sangaku. To proceed, we need to fill some of the empty cells in the table above:

DG / AC = BD / AB,

so that DG = (1 - k)ab/c. Since DE = kc, the area S of the rectangle DEFG is given by

S = DG × DE = (1 - k)k·ab

which attains its maximum along with

## 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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Copyright © 1996-2018 Alexander Bogomolny

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