Sangaku and The Egyptian Triangle
Three sangaku that require to determine the relative radii of the circles shown can be solved by a direct application of the Pythagorean theorem, see below, but if combined into a single (and a little amplified, sangakuready) configuration
reveal existence of possible relationships not obvious when they are studied separately. The configuration in fact, as was noted by L. Bankoff and C. W. Trigg quarter of a century before sangaku grew in popularity after a 1998 Scientific American article, is rich with surprises, the main being the numerous sightings of the famous 3:4:5 triangle.
The latter is most often referred to as the Ropestretchers triangle and sometimes as the Egyptian triangle. (Both because of the belief that this simplest of the Pythagorean triangles was used by the ancient ropestretchers in construction and, in particular, in construction of the Great Pyramids. Sometimes, however, the term Egyptian triangle is preserved for the one related to the dimensions of the pyramid of Cheops and the golden ratio.)
For convenience, assume the side of the square is 24 so that
HC = HG as tangents to (F) from H. For the same reason
Then HB = 18, AH = 30, AK = 15, KG = 9,
We see that triangles FCJ, HGJ, FGK, AKE, AEI, AEM, and ABH are all 3:4:5 triangles.
Let T be the intersection of AI extended with (A). Then
RU = FI  SR = 8  16/3 = 8/3 = RT. 
In addition, if V is on BC and VC = RU, then
References
 L. Bankoff, C. W. Trigg, The Ubiquitous 3:4:5 Triangle, Math Magazine, v 47, n 2 (Mar., 1974), pp. 6170
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

[an error occurred while processing this directive]
Contact Front page Contents Geometry Up
Copyright © 19962018 Alexander Bogomolny
Let's, for example find the radius of (K) tangent to (A), (B) and AB. Actually we already saw that
(24  r)^{2}  r^{2} = 12^{2}, 
which gives
576  48r = 144, or 48r = 432, r = 9. 
Contact Front page Contents Geometry Up
Copyright © 19962018 Alexander Bogomolny