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Equilateral Triangle, Straight Line and Tangent Circles: What Is This About?
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Copyright © 1996-2010 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
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(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
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x = AM = AK, y = BN = BL. |
Then
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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
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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Copyright © 1996-2010 Alexander Bogomolny
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