Equilateral Triangle, Straight Line and Tangent Circles: What Is This About?
A Mathematical Droodle

 

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Solution

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Copyright © 1996-2017 Alexander Bogomolny

The applet purports to suggest the following sangaku [Temple Geometry, p. 25, #2.1.11]:

  ABC is equilateral triangle, and l is any line through vertex C. The circle O(ra) touches l, AC and AB, and the circle Q(rb) touches l, BC, and AB. Show that as the line l varies the sum ra + rb remains constant. Furthermore, if h is the altitude of ΔABC, then

  ra + rb = h.

 

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?

(This is a 1885 Sangaku from the Fukusima prefecture whose tablet has disappeared long ago.)

Let K, L, M, N, S, T be the points of tangency as shown in the applet and s be the common length of the sides of ΔABC. Denote

  x = AM = AK,
y = BN = BL.

Then

  s - x = CM = CS,
s - y = CN = CT.

Two external tangents to a pair of circles are equal: KL = ST which tells us that

  2s - (x + y) = ST = KL = (x + y) + s,

implying

  x + y = s/2.

But triangles OAK and QBL are both 30°-60°-90°. Therefore, ra = 3·x and rb = 3·y such that

  ra + rb = 3/2·s = h.

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

Sangaku


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