Pythagoras and Vecten Break Japan's Isolation

Here we have a third solution to a Sangaku problem: Five squares are arranged as in the applet. Show that the area of triangle KMN equals the area of the square BEKH. The other two can be found elsewhere. The solution has been suggested by Nathan Bowler; it does not invoke any algebraic means.

The solution is based on two key observations:

• First, as in Solution #2, we observe that there are pairs of flank triangles in Vecten configuration. Thus we see that the areas of the four triangles ABE, DEK, BCH, KHI are equal, as are the areas of triangles KMN and DIK.

• B is the midpoint of XY. Thus, when square BEKH is cut into 5 pieces by vertical lines through B and K and horizontal lines through E and H, in the manner of Proof #3 of the Pythagorean theorem, points U and V being directly above B, the quadrilateral DUIV is a parallelogram (with center O.) So that triangles DOU and IOV have equal areas, from which

Now, taking into account the first observation, we see that

which implies the required identity

The last step admits a very slight modification that invokes Proof #10 of the Pythagorean theorem. Indeed, for the same reasons as above,

Sangaku

1. Sangaku: Reflections on the Phenomenon
2. Critique of My View and a Response
3. 1 + 27 = 12 + 16 Sangaku
4. 3-4-5 Triangle by a Kid
5. 7 = 2 + 5 Sangaku
6. A 49^th Degree Challenge
7. A Geometric Mean Sangaku
8. A Hard but Important Sangaku
9. A Sangaku: Two Unrelated Circles
10. A Sangaku by a Teen
11. A Sangaku Follow-Up on an Archimedes' Lemma
12. A Sangaku with an Egyptian Attachment
13. A Sangaku with Many Circles and Some
14. An Old Japanese Theorem
15. Archimedes Twins in the Edo Period
16. Arithmetic Mean Sangaku
17. Bottema Shatters Japan's Seclusion
18. Circles and Semicircles in Rectangle
19. Circles in a Circular Segment
20. Circles Lined on the Legs of a Right Triangle
21. Equal Incircles Theorem
22. Equilateral Triangle, Straight Line and Tangent Circles
23. Equilateral Triangles and Incircles in a Square
24. Five Incircles in a Square
25. Four Hinged Squares
26. Four Incircles in Equilateral Triangle
27. Gion Shrine Problem
28. Harmonic Mean Sangaku
29. Heron's Problem
30. In the Wasan Spirit
31. Incenters in Cyclic Quadrilateral
32. Japanese Art and Mathematics
33. Malfatti's Problem
34. Maximal Properties of the Pythagorean Relation
35. Neuberg Sangaku
36. Out of Pentagon Sangaku
37. Peacock Tail Sangaku
38. Pentagon Proportions Sangaku
39. Pythagoras and Vecten Break Japan's Isolation
40. Radius of a Circle by Paper Folding
41. Review of Sacred Mathematics
42. Sangaku ŕ la V. Thebault
43. Sangaku and The Egyptian Triangle
44. Sangaku in a Square
45. Sangaku Iterations, Is it Wasan?
46. Sangaku with 8 Circles
47. Sangaku with Three Mixtilinear Circles
48. Sangaku with Versines
49. Sangakus with a Mixtilinear Circle
50. Sequences of Touching Circles
51. Square and Circle in a Gothic Cupola
52. Tangent Circles and an Isosceles Triangle
53. The Squinting Eyes Theorem
54. Steiner's Sangaku
55. Three Incircles In a Right Triangle
56. Three Squares and Two Ellipses
57. Three Tangent Circles Sangaku
58. Triangles, Squares and Areas from Temple Geometry
59. Two Arbelos, Two Chains
60. Two Circles in an Angle

References

1. J. Konhauser, D. Velleman, S. Wagon, Which Way Did the Bicycle Go?, MAA, 1996, #50

Copyright © 1996-2008 Alexander Bogomolny