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

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Solution

## Sangaku

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