Three Incircles In a Right Triangle: What Is This About?
A Mathematical Droodle

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Solution

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Copyright © 1996-2018 Alexander Bogomolny

The applet purports to suggest the following sangaku [Temple Geometry, #2.3.2, p. 29]:

ABC is right-angled at C, and CD is the perpendicular from C onto AB. If O1(r1), O2(r2), O3(r3) are the incircles of the respective triangles ABC, ADC, and BDC, show that   r1 + r2 + r3 = CD.

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(This is an undated Sangaku from the Iwate prefecture.)

The three triangles are right-angled and, therefore, similar. Let the sides of ΔABC be a, b, c, in the usual manner. Then corresponding sides of ΔADC are b²/2, ab/c, b, and those of ΔBDC are ab/c, a²/c, and a. The inradii of the three triangles are easily found to be

(1)	r1 = (a + b - c)/2 = c/c·(a + b - c)/2, r2 = (b²/2 + ab/c - b)/2 = b/c·(a + b - c)/2, r3 = (ab/c + a²/c - a)/2 = a/c·(a + b - c)/2.

Adding the three identities gives

(2)	r1 + r2 + r3 = (a + b - c)(a + b + c) / 2c   = [(a + b)² - c²] / 2c   = [(a² + b² - c²) + 2ab] / 2c   = ab / c,

where we have used the Pythagorean theorem. For the area S of ΔABC we have 2S = ab, on the one hand, and, on the other, 2S = ch, where h is the altitude CD. Comparing the two shows that ab/c = h, as required.

Note that the derivation (1)-(2) may be seen as the converse of one of the proofs of the Pythagorean theorem.

References

1. H. Fukagawa, D. Pedoe, Japanese Temple Geometry Problems, The Charles Babbage Research Center, Winnipeg, 1989

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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 Restored Sangaku Problem
   • A better solution to a difficult sangaku problem
   • A Simple Solution to a Difficult Sangaku Problem
   • A Trigonometric Solution to a Difficult Sangaku Problem
10. A Sangaku: Two Unrelated Circles
11. A Sangaku by a Teen
12. A Sangaku Follow-Up on an Archimedes' Lemma
13. A Sangaku with an Egyptian Attachment
14. A Sangaku with Many Circles and Some
15. A Sushi Morsel
16. An Old Japanese Theorem
17. Archimedes Twins in the Edo Period
18. Arithmetic Mean Sangaku
19. Bottema Shatters Japan's Seclusion
20. Chain of Circles on a Chord
21. Circles and Semicircles in Rectangle
22. Circles in a Circular Segment
23. Circles Lined on the Legs of a Right Triangle
24. Equal Incircles Theorem
    • Equal Incircle Theorem, Angela Drei's Proof
25. Equilateral Triangle, Straight Line and Tangent Circles
26. Equilateral Triangles and Incircles in a Square
27. Five Incircles in a Square
28. Four Hinged Squares
29. Four Incircles in Equilateral Triangle
    • Four Incircles in an Equilateral Triangle, a Sangaku
30. Gion Shrine Problem
31. Harmonic Mean Sangaku
32. Heron's Problem
33. In the Wasan Spirit
34. Incenters in Cyclic Quadrilateral
35. Japanese Art and Mathematics
36. Malfatti's Problem
37. Maximal Properties of the Pythagorean Relation
38. Neuberg Sangaku
39. Out of Pentagon Sangaku
40. Peacock Tail Sangaku
41. Pentagon Proportions Sangaku
42. Proportions in Square
43. Pythagoras and Vecten Break Japan's Isolation
44. Radius of a Circle by Paper Folding
45. Review of Sacred Mathematics
46. Sangaku à la V. Thebault
47. Sangaku and The Egyptian Triangle
48. Sangaku in a Square
49. Sangaku Iterations, Is it Wasan?
50. Sangaku with 8 Circles
51. Sangaku with Angle between a Tangent and a Chord
52. Sangaku with Quadratic Optimization
53. Sangaku with Three Mixtilinear Circles
54. Sangaku with Versines
55. Sangakus with a Mixtilinear Circle
56. Sequences of Touching Circles
57. Square and Circle in a Gothic Cupola
58. Steiner's Sangaku
59. Tangent Circles and an Isosceles Triangle
60. The Squinting Eyes Theorem
61. Three Incircles In a Right Triangle
62. Three Squares and Two Ellipses
63. Three Tangent Circles Sangaku
    • An Extension of a Sangaku with Touching Circles
64. Triangles, Squares and Areas from Temple Geometry
65. Two Arbelos, Two Chains
66. Two Circles in an Angle
67. Two Sangaku with Equal Incircles
68. Another Sangaku in Square
69. Sangaku via Peru
70. FJG Capitan's Sangaku

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Copyright © 1996-2018 Alexander Bogomolny