Sangakus with a Mixtilinear Circle
What Is This About?
A Mathematical Droodle


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The applet purports to suggest the following sangaku [Temple Geometry, 2.3.3]:

Triangle ABC is inscribed in the circle O(R) and AB is a diameter. The circle O1(r1) touches CA and CB and touches O(R) internally, and I(r) is the incircle of triangle ABC. Show that

r1 = 2r.

(This is a 1842 Sangaku from the Iwate prefecture. Another sangaku (2.2.7) does not mention the incircle but requests a proof of r1 = a + b - c, where a, b are the legs and c the hypotenuse of ΔABC. This one was written in 1893, 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 formulation of the second sangaku suggests there is a direct way to calculate r1. I do not see it at this point. The problem is solved with a reference to a more general case:

r = r1 cos2(α/2).

Since in the present problems α = 90°, cos(α/2) = 2/2 and, as a consequence, r = r1/2, as required.

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

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  • Sangaku with Three Mixtilinear Circles
  • Construction and Properties of Mixtilinear Incircles
  • Construction and Properties of Mixtilinear Incircles 2
  • Conic in Mixtilinear Incircles
  • A Sangaku: Two Unrelated Circles
  • Sangaku

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