# Sangaku in a Square

One of the simplest sangaku - geometric problems carved on colorful wooden tablets and offered in shinto shrines and buddhist temples during the Edo period (1603-1867) of self-imposed seclusion of Japan from the Western world - is presented by the following diagram:

A triangle is formed by a line that joins the base of a square with the midpoint of the opposite side and a diagonal. Find the radius of the inscribed circle.

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Copyright © 1996-2018 Alexander Bogomolny

A triangle is formed by two lines that join the base of a square with the midpoint of the opposite side and a diagonal. Find the radius of the inscribed circle.

A solution could be observed on an extended diagram. Let A, B, C, D be the vertices of the square and M the midpoint of AD. BD and CM intersect in P. Let Q be the intersection of BM and CD and H the projection of P on BC. The task is to find the inradius of ΔBCP.

Focus on ΔBCQ. MD is parallel to the base and is half as long which implies that this is a midline of the triangle. In other words, M and D are the midpoints of BQ and CQ, respectively. This means that BD and CM are two medians in the triangle and P its centroid. The centroid divides the medians in ratio 2:1 so that

(1)

CP = CM·2/3 and BP = BD·2/3.

Thus assuming BC = a, we can apply the Pythagorean theorem to first find BD (from ΔBCD) and CM (from ΔCDM) and subsequently, from (1) the sides of ΔBCP. By the same token, altitude

(2)

r·p = 2S,

where r is the inradius, p the perimeter, and S the area of the triangle. Putting everything together we see that

r·a·(1 + √5/2·2/3 + √2·2/3) = 2·a^{2}/3,

from which r is easily found:

(3)

r = 2·a / (3 + √5 + 2√2).

The problem perhaps warrants a

### Remark

The problem is clearly computational: the question is to find the inradius of a triangle related to a given square. Students may be tempted to use calculators or even dynamic geometry software to avoid handling square roots and fractions. This is what they might get - more or less. Assume

BD = 1.414213562,

BP = .9428090416,

CM = 1.118033989,

CP = .7453559925,

p = 2.688165034,

QH = .6666666667,

S = .3333333333,

2S = .6666666666,

r = .2480006466,

which is quite in agreement with the exact answer (3). The difference between the two is that the numeric value, however accurate, carries no information as to the manner in which it was obtained. In (3), the appearance of the square roots is suggestive and points to a possible application of the Pythagorean theorem to two right triangles, as we did above. Thus (3) exhibits a valuable pattern that may trigger a thought process that may lead to the recollection of properties of medians and centroids, whereas

In short, this 300 year old problem serves an example where the use of calculators actually obscures relationships between numbers. More such examples can be found in an edifying book *What The Numbers Say* by D. Niederman and D. Boyum along with a multitude of observations on everyday number usage and the ability to see beyond the numbers.

### 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 1L6D. Niederman and D. Boyum,

*What The Numbers Say*, Broadway Books, 2003

## 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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