## 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.

What if applet does not run? |

|Activities| |Contact| |Front page| |Contents| |Geometry|

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.

What if applet does not run? |

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

- M and N are the points of tangency of (O) with the angle,
- (Q) is tangent to MN,
- 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

### 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:

### 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:

- Sangaku: Reflections on the Phenomenon
- Critique of My View and a Response
- 1 + 27 = 12 + 16 Sangaku
- 3-4-5 Triangle by a Kid
- 7 = 2 + 5 Sangaku
- A 49
^{th}Degree Challenge - A Geometric Mean Sangaku
- A Hard but Important Sangaku
- A Restored Sangaku Problem
- A Sangaku: Two Unrelated Circles
- A Sangaku by a Teen
- A Sangaku Follow-Up on an Archimedes' Lemma
- A Sangaku with an Egyptian Attachment
- A Sangaku with Many Circles and Some
- A Sushi Morsel
- An Old Japanese Theorem
- Archimedes Twins in the Edo Period
- Arithmetic Mean Sangaku
- Bottema Shatters Japan's Seclusion
- Chain of Circles on a Chord
- Circles and Semicircles in Rectangle
- Circles in a Circular Segment
- Circles Lined on the Legs of a Right Triangle
- Equal Incircles Theorem
- Equilateral Triangle, Straight Line and Tangent Circles
- Equilateral Triangles and Incircles in a Square
- Five Incircles in a Square
- Four Hinged Squares
- Four Incircles in Equilateral Triangle
- Gion Shrine Problem
- Harmonic Mean Sangaku
- Heron's Problem
- In the Wasan Spirit
- Incenters in Cyclic Quadrilateral
- Japanese Art and Mathematics
- Malfatti's Problem
- Maximal Properties of the Pythagorean Relation
- Neuberg Sangaku
- Out of Pentagon Sangaku
- Peacock Tail Sangaku
- Pentagon Proportions Sangaku
- Proportions in Square
- Pythagoras and Vecten Break Japan's Isolation
- Radius of a Circle by Paper Folding
- Review of Sacred Mathematics
- Sangaku à la V. Thebault
- Sangaku and The Egyptian Triangle
- Sangaku in a Square
- Sangaku Iterations, Is it Wasan?
- Sangaku with 8 Circles
- Sangaku with Angle between a Tangent and a Chord
- Sangaku with Quadratic Optimization
- Sangaku with Three Mixtilinear Circles
- Sangaku with Versines
- Sangakus with a Mixtilinear Circle
- Sequences of Touching Circles
- Square and Circle in a Gothic Cupola
- Steiner's Sangaku
- Tangent Circles and an Isosceles Triangle
- The Squinting Eyes Theorem
- Three Incircles In a Right Triangle
- Three Squares and Two Ellipses
- Three Tangent Circles Sangaku
- Triangles, Squares and Areas from Temple Geometry
- Two Arbelos, Two Chains
- Two Circles in an Angle
- Two Sangaku with Equal Incircles
- Another Sangaku in Square
- Sangaku via Peru
- FJG Capitan's Sangaku

### References

H. Fukagawa, D. Pedoe,

*Japanese Temple Geometry Problems*, The Charles Babbage Research Center, Winnipeg, 1989Write to:

Charles Babbage Research Center

P.O. Box 272, St. Norbert Postal Station

Winnipeg, MB

Canada R3V 1L6

|Activities| |Contact| |Front page| |Contents| |Geometry|

Copyright © 1996-2006 Alexander Bogomolny

67011222