Three Incircles In a Right Triangle: What Is This About?
A Mathematical Droodle

What if applet does not run? 
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny
The applet purports to suggest the following sangaku [Temple Geometry, #2.3.2, p. 29]:
ABC is rightangled at C, and CD is the perpendicular from C onto AB. If O_{1}(r_{1}), O_{2}(r_{2}), O_{3}(r_{3}) are the incircles of the respective triangles ABC, ADC, and BDC, show that


What if applet does not run? 
(This is an undated Sangaku from the Iwate prefecture.)
The three triangles are rightangled and, therefore, similar. Let the sides of ΔABC be a, b, c, in the usual manner. Then corresponding sides of ΔADC are b²/2, ab/c, b, and those of ΔBDC are ab/c, a²/c, and a. The inradii of the three triangles are easily found to be
(1) 
r_{1} = (a + b  c)/2 = c/c·(a + b  c)/2, r_{2} = (b²/2 + ab/c  b)/2 = b/c·(a + b  c)/2, r_{3} = (ab/c + a²/c  a)/2 = a/c·(a + b  c)/2. 
Adding the three identities gives
(2) 

where we have used the Pythagorean theorem. For the area S of ΔABC we have
Note that the derivation (1)(2) may be seen as the converse of one of the proofs of the Pythagorean theorem.
References
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
Sangaku
 Sangaku: Reflections on the Phenomenon
 Critique of My View and a Response
 1 + 27 = 12 + 16 Sangaku
 345 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 FollowUp 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
Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny
68223247