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

14 December 2015, Created with GeoGebra

Explanation

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Copyright © 1996-2017 Alexander Bogomolny

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.

Two points A and B are given on the same side of a line l. Find a point M on l such that the sum of distances from A and B to M is minimal.

Heron's problem

Think of l as a mirror. Heron posed that the shortest distance between A and B via l is exactly the path traversed by a ray of light emitted from A and observed at B. From here he deduced that when light is reflected in a mirror the angle of incidence fi is equal to the angle of reflection fr, where the angles at hand are formed by AM and MB with the perpendicular to l at M.

Why so?

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 ( ΔAMB'),

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:

P is a variable point in the given segment AB, and C is a fixed point on the perpendicular AC to the line AB, and D is fixed, on the same side of AB as C, and lies on the perpendicular BD to AB. AC = c, BD = d, and AB = k. Find AP = p such that CP + PD is a minimum when P moves on AB.

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

AP / AC = BP /BD = (AP + BP) / (AC + BD),

or,

p / c = k / (c + d),

so that

p = kc / (c + d).

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 49th Degree Challenge
  7. A Geometric Mean Sangaku
  8. A Hard but Important Sangaku
  9. A Restored 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
  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
  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
  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

Related material
Read more...

  • Heron's Formula: a Proof
  • A Proof of the Pythagorean Theorem From Heron's Formula
  • A Proof of the Pythagorean Theorem From Heron's Formula II
  • A few corollaries from the Pythagorean Theorem

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