Dynamic Geometry 1452: Japanese Theorem, Sangaku

Given a cyclic quadrilateral ABCD and let A1, B1, C1, D1 the incenters and ra, rb, rc, rd the inradii of the triangles BAD, ABC, BCD, ADC. Then (1) A1B1C1,D1 is a rectangle, (2) ra + rc = rb + rd

This sangaku problem was proposed by Maruyama Ryoukan in 1800. References: Fukagawa Hidetoshi, Tony Rothman, "Sacred Mathematics: Japanese Temple Geometry" (Princenton University Press, 2008).

See solution below

Static Diagram of the Japanese Theorem, Sangaku

Dynamic Geometry 1452: Japanese Theorem, Sangaku. Using GeoGebra

Poster of the Japanese Theorem, Sangaku using iPad Apps

Poster of Japanese Theorem, Sangaku, Dynamic Geometry. Using iPad

Search gogeometry.com

Classroom Resource:
Interactive step-by-step animation using GeoGebra

This step-by-step interactive illustration was created with GeoGebra.

  • To explore (show / hide): click/tap a check box.
  • To stop/play the animation: click/tap the icon in the lower left corner.
  • To go to first step: click/tap the "Go to step 1" button.
  • To manipulate the interactive figure: click/tap and drag the points or figures.

GeoGebra is free and multi-platform dynamic mathematics software for all levels of education that joins geometry, algebra, tables, graphing, statistics and calculus application, intended for teachers and students. Many parts of GeoGebra have been ported to HTML5.

Recent Additions

Geometry Problems
Open Problems
Visual Index
Ten problems: 1411-1420
All Problems
Cyclic Quadrilateral
Classical Theorems
Dynamic Geometry
HTML5 and Dynamic Geometry
iPad Apps
View or Post a solution

Geometry Problem 1452 Solution(s)