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


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Copyright © 1996-2017 Alexander Bogomolny

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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    39. Out of Pentagon Sangaku
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    43. Pythagoras and Vecten Break Japan's Isolation
    44. Radius of a Circle by Paper Folding
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    47. Sangaku and The Egyptian Triangle
    48. Sangaku in a Square
    49. Sangaku Iterations, Is it Wasan?
    50. Sangaku with 8 Circles
    51. Sangaku with Angle between a Tangent and a Chord
    52. Sangaku with Quadratic Optimization
    53. Sangaku with Three Mixtilinear Circles
    54. Sangaku with Versines
    55. Sangakus with a Mixtilinear Circle
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