Sangaku à la V. Thébault: What Is This About?
A Mathematical Droodle


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

Solution

Sangaku

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

Copyright © 1996-2017 Alexander Bogomolny

The applet is hopefully suggestive of a generalization of the following sangaku [Fukagawa & Pedoe, Example 2.4]:

O(R) is the circumcircle and I(r) is the incircle of triangle ABC, K is the foot of the perpendicular from C to AB. The circle P(r1) touches CK and AB and touches O(R) internally, and Q(r2) touches CK and AB and touches O(R) internally. Show that

r1 + r2 = 2r.

The problem has been written on a surviving tablet in 1901, in the Fukusima prefecture.


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

The applet suggests more than claimed: in fact I is the midpoint of the segment PQ. (Since all three circles touch the segment AB, the latter fact implies r1 + r2 = 2r. That the three points are collinear follows from one of V. Thébault's problems.)

The proof follows along the lines of another sangaku.

References

  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

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

Copyright © 1996-2017 Alexander Bogomolny

 62636900

Search by google: