FJG Capitán's Sangaku


FJG Capitán's Sangaku


Assume that the left yellow circle is described by $x^2+(y-R_1)^2=R_1^2;\,$ the right yellow circle by $(x-a)^2+(y-R_2)^2=R_2^2,\,$ $a\gt 0.\,$ The two being externally tangent, we have by the Pythagorean theorem,


implying $a=2\sqrt{R_1R_2}.\,$ Hence, the right yellow circle is described by $(x-2\sqrt{R_1R_2})^2+(y-R_2)^2=R_2^2.\,$

The rightmost red circle is described by $(x-b)^2+(y-r)^2=r^2.\,$ It being externally tangent to the right yellow circle, we get $(2\sqrt{R_1R_2}-b)^2+(r-R_2)^2=(r+R_2)^2,\,$ implying $b=2\sqrt{R_1R_2}-2\sqrt{rR_2}.$

Similarly, the leftmost red circle has the equation $(x-c)^2+(y-r)^2=r^2,\,$ where $c=2\sqrt{rR_1}.\,$ But $b-c=2(n-1)r\,$ such that $(n-1)r=\sqrt{R_1R_2}-\sqrt{r}(\sqrt{R_1}+\sqrt{R_2}),\,$ i.e., $\sqrt{r}(\sqrt{R_1}+\sqrt{R_2})=\sqrt{R_1R_2}-(n-1)r,\,$ and, by squaring, we obtain



The above problem has been kindly posted by Leo Giugiuc, along with his solution at the CutTheKnotMath facebook page. The problem - by Francisco Javier García Capitán - was originally posted at the Οι Ρομαντικοι της Γεωμετριας (Romantics of Geometry) facebook group.


Related material

  • 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 49th Degree Challenge
  • A Geometric Mean Sangaku
  • A Hard but Important Sangaku
  • A Restored Sangaku Problem
  • A better solution to a difficult sangaku problem
  • A Simple Solution to a Difficult Sangaku Problem
  • A Trigonometric Solution to a Difficult 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
  • Equal Incircle Theorem, Angela Drei's Proof
  • 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
  • Four Incircles in an Equilateral Triangle, a Sangaku
  • 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
  • An Extension of a Sangaku with Touching Circles
  • 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
  • |Contact| |Front page| |Contents| |Geometry|

    Copyright © 1996-2018 Alexander Bogomolny