Two Circles in an Angle: What is this about?
A Mathematical Droodle

A simple sangaku deals with two circles inside an angle. The circles are related in two ways which imply each other.

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Copyright © 1996-2006 Alexander Bogomolny

The applet presents two circles inscribed in an angle. One of the circles is incident with the center of the other. The latter is tangent to the chord in the former that joins the points of tangency with the angle.

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More accurately, let there be two circles - (O) centered at O, (Q) centered at Q - inscribed into an angle with vertex C. (Obviously, OQC is the bisector of the angle.) Let MN be a chord in (O) perpendicular to OC. Then any two of the three conditions

  1. M and N are the points of tangency of (O) with the angle,
  2. (Q) is tangent to MN,
  3. Q lies on (O)

imply the third. There are two cases depending on whether O is father from C than Q or nearer. I shall treat only the former.

1 and 2 imply 3

Let Q' and V be the points where OC meets (O), V farthest from C. The tangent to (O) at V meets CM and CN in U and W, respectively such that (O) is inscribed into ΔCUW. O then is the incenter of that triangle and is located at the intersection of its internal angle bisectors. It follows that

(1) ∠CUO = ∠OUV.

In addition, quadrilateral OMUV is cyclic (because both angles OMU and OVU are right) which shows that

(2) ∠MUV = 180° - ∠OUV = ∠MOQ'.

The latter is the central angle in (O) subtended by chord MQ' and thus is twice the angle formed by the chord and a tangent at one of its ends:

(3) ∠MOQ' = 2·∠CMQ'.

From (1)-(3),

(4) ∠CUO = 2·∠CMQ',

UO||MQ'. Since also UW||MN, this means that MQ' is the bisector of ∠CMN in ΔCMN. Q' is then the incenter of that triangle. Since (Q) is tangent to all three side lines of the latter, so is Q. We conclude that Q' = Q.

1 and 3 imply 2

Let (Q') be the incircle of ΔCMN. As we just seen this assumption implies that Q' lies on (O). Since the same holds for Q and both are on the same side from O as C, they coincide: Q = Q'. (Q) is the same as (Q') and is therefore tangent to MN.

2 and 3 imply 1

Let M' and N' be the points where (O) touches CU and CV, respectively. Let O(Q') be the incircle of ΔCM'N'. Then, as we already observed, Q' is bound to lie on (O); it thus coincides with Q: Q = Q'.

  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


  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

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Copyright © 1996-2006 Alexander Bogomolny