Malfatti's Problem

Malfatti's problem serves a miniature example of broader, often torturous, mathematical evolution. It has started with an apparently practical problem: to cut out of a triangular prism of a material such as marble, three circular columns of the greatest possible volume. The problem (now dubbed the marble problem [Martin, p.92]) has been published in 1803 by G. Malfatti. The problem is naturally reduced to a planar problem of packing into a given triangle three circles of the largest total area. In his solution, Malfatti assumed that the three circles in the marble problem must touch each other and each touch two sides of the triangle. Thus he reduced the problem to the one that now bears his name, Malfatti's problem [Dörrie, 147]:

It is said that a century earlier Jacob Bernoulli considered the problem for an isosceles triangle, which, if true, would have enhanced its standing in the mathematical community. (The problem also occurs in Japanese temple geometry where it is often attributed to Chokuyen Naonobu Ajima, 1732-1798.) The problem received a serious boost in 1826 when J. Steiner published (without a proof!) an elegant solution, probably with a purpose of demonstrating the power of his methods. Following is a short history of further development [Bottema]:

When the three circles in Steiner's generalization are pairwise tangent externally, the problem is trivialized by inversion [Bottema]. An inversion with center in one of the points of tangency, inverts two circles into two parallel lines:

where C₁, C₂, C₃ are the inversions of the given circles, while M₁, M₂, M₃ invert back into Malfatti's circles for the generalized problem.

Of the texts available to me and listed below, Dörrie gives a mostly analytic but otherwise an elementary solution by Schellbach. Martin seems to follow a similar solution by Malfatti himself. Coolidge, F. G.-M., Hadamard, Shklyarsky all follow in Steiner's footsteps, but with different proofs. The latter does not mention either Malfatti or Steiner. Wells describes the problem without proof, gives a short historic outline and provides counterexamples.

1. Draw three angle bisectors IA, IB, IC.

2. In the triangles IAB, IBC, ICA inscribe circles C_c, C_a, C_b. Note that the angle bisectors serve as common internal tangents for pairs of these circles.

3. For each pair of the circles consider the second internal tangents. The latter concur in a point (L) and cross the sides in points M, P, T, as shown in the applet.

4. The three quadrilaterals APLT, BMLP, and CTLM are inscriptible. Their incircles solve Malfatti's problem.

The proof appears elsewhere.

References

1. O. Bottema, The Malfatti Problem, Forum Geometricorum, v 1 (2000), pp. 43-50 (Ttranslation by F. van Lamoen of an 1949 article)
2. J. L. Coolidge, A Treatise On the Circle and the Sphere, AMS - Chelsea Publishing, 1971
3. H. Dörrie, 100 Great Problems Of Elementary Mathematics, Dover Publications, NY, 1965.
4. F. G.-M., Exercices de Géométrie, Éditions Jacques Gabay, sixiéme édition, 1991, pp. 726-728
5. H. Fukagawa, D. Pedoe, Japanese Temple Geometry Problems, The Charles Babbage Research Center, Winnipeg, 1989
6. M. Goldberg, On the Original Malfatti Problem, Math. Mag. 40, 241-247, 1967.
7. J. Hadamard, Leçons de géométrie élémentaire, tome I, 13e édition, reprint 1988
8. G. E. Martin, A Geometric Constructions, Springer, 1998
9. D. O. Shklyarsky, N. N. Chentsov, I. M. Yaglom, Selected Problems and Theorems of Elementary Mathematics, v 2, Moscow, 1952 (in Russian)
10. D. Wells, The Penguin Book of Curious and Interesting Puzzles, Penguin Books, 1992

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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
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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. An Old Japanese Theorem
16. Archimedes Twins in the Edo Period
17. Arithmetic Mean Sangaku
18. Bottema Shatters Japan's Seclusion
19. Circles and Semicircles in Rectangle
20. Circles in a Circular Segment
21. Circles Lined on the Legs of a Right Triangle
22. Equal Incircles Theorem
23. Equilateral Triangle, Straight Line and Tangent Circles
24. Equilateral Triangles and Incircles in a Square
25. Five Incircles in a Square
26. Four Hinged Squares
27. Four Incircles in Equilateral Triangle
28. Gion Shrine Problem
29. Harmonic Mean Sangaku
30. Heron's Problem
31. In the Wasan Spirit
32. Incenters in Cyclic Quadrilateral
33. Japanese Art and Mathematics
34. Malfatti's Problem
35. Maximal Properties of the Pythagorean Relation
36. Neuberg Sangaku
37. Out of Pentagon Sangaku
38. Peacock Tail Sangaku
39. Pentagon Proportions Sangaku
40. Pythagoras and Vecten Break Japan's Isolation
41. Radius of a Circle by Paper Folding
42. Review of Sacred Mathematics
43. Sangaku à la V. Thebault
44. Sangaku and The Egyptian Triangle
45. Sangaku in a Square
46. Sangaku Iterations, Is it Wasan?
47. Sangaku with 8 Circles
48. Sangaku with Three Mixtilinear Circles
49. Sangaku with Versines
50. Sangakus with a Mixtilinear Circle
51. Sequences of Touching Circles
52. Square and Circle in a Gothic Cupola
53. Tangent Circles and an Isosceles Triangle
54. The Squinting Eyes Theorem
55. Steiner's Sangaku
56. Three Incircles In a Right Triangle
57. Three Squares and Two Ellipses
58. Three Tangent Circles Sangaku
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61. Two Circles in an Angle
62. Two Sangaku with Equal Incircles