Equilateral Triangle, Straight Line and Tangent Circles: What Is This About?
A Mathematical Droodle
What if applet does not run? 
Activities Contact Front page Contents Geometry
Copyright © 19962018 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(r_{a}) touches l, AC and AB, and the circle Q(r_{b}) touches l, BC, and AB. Show that as the line l varies the sum

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, r_{a} = √3·x and r_{b} = √3·y such that
r_{a} + r_{b} = √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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Activities Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny