Three Incircles In a Right Triangle: What Is This About?
A Mathematical Droodle

What if applet does not run? 
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Copyright © 19962017 Alexander Bogomolny
The applet purports to suggest the following sangaku [Temple Geometry, #2.3.2, p. 29]:
ABC is rightangled at C, and CD is the perpendicular from C onto AB. If O_{1}(r_{1}), O_{2}(r_{2}), O_{3}(r_{3}) are the incircles of the respective triangles ABC, ADC, and BDC, show that


What if applet does not run? 
(This is an undated Sangaku from the Iwate prefecture.)
The three triangles are rightangled and, therefore, similar. Let the sides of ΔABC be a, b, c, in the usual manner. Then corresponding sides of ΔADC are b²/2, ab/c, b, and those of ΔBDC are ab/c, a²/c, and a. The inradii of the three triangles are easily found to be
(1) 
r_{1} = (a + b  c)/2 = c/c·(a + b  c)/2, r_{2} = (b²/2 + ab/c  b)/2 = b/c·(a + b  c)/2, r_{3} = (ab/c + a²/c  a)/2 = a/c·(a + b  c)/2. 
Adding the three identities gives
(2) 

where we have used the Pythagorean theorem. For the area S of ΔABC we have
Note that the derivation (1)(2) may be seen as the converse of one of the proofs of the Pythagorean theorem.
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
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Copyright © 19962017 Alexander Bogomolny
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