Heron's Problem: What is it?
A Mathematical Droodle

Explanation

Heron's Problem

This one is a basic optimization problem. It's quite famous, being discussed in Heron's Catoptrica (On Mirrors from the Greek word Katoptron Catoptron = Mirror) that, in all likelihood, saw the light of day some 2000 years ago.

Let B' be the symmetric image of B in l, such that BB' is perpendicular to l and is divided by the latter in halves. MB = MB', for any point M on l. Therefore, AM + MB = AM + MB'. By the triangle inequality (AMB'),

AM + MB' AB'.

with the equality reached only when M lies on AB', in which case clearly the two angles coincide.

Heron's problem has applications to curved surfaces, ellipse for one.

Note: the problem has been known in Japan. It appeared on a now lost sangaku tablet from 1830 written in the Yamagata prefecture [Fukagawa and Pedoe, problem 4.1.1]. The Japanese variant underscored the computational aspects of the solution:

It is not difficult to surmise the solution from the diagram

When P satisfies the minimality requirement, triangles APC and BPD' are similar from which

or,

so that

References

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

   Write to:

   Charles Babbage Research Center
   P.O. Box 272, St. Norbert Postal Station
   Winnipeg, MB
   Canada R3V 1L6

2. P. J. Nahin, When Least Is Best, Princeton University Press, 2007 (Fifth printing).
3. V. M. Tikhomirov, Stories about Maximua and Minima, AMS & MAA, 1990
4. D. Wells, The Penguin Book of Curious and Interesting Geometry, Penguin, 1991
5. I. M. Yaglom, Geometric Transformations I, MAA, 1962

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