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

 

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Explanation

Copyright © 1996-2008 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.

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 (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:

  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. V. M. Tikhomirov, Stories about Maximua and Minima, AMS & MAA, 1990
  3. D. Wells, The Penguin Book of Curious and Interesting Geometry, Penguin, 1991
  4. 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 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

Copyright © 1996-2008 Alexander Bogomolny

28677888Page copy protected against web site content infringement by Copyscape


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