Right-angled triangle and incircles

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mihiv
Thank you for the elegant solution! I saw 6 or 7 different solutions to this problem and all of them was totally different. I suppose there are even more. It is the reason here http://www.cut-the-knot.org/m/Geometry/ ... ngle.shtml to have just two solutions.

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Очевидно, красная и синяя площади равны. Но это и означает, что $(r_1+r_3)^2+(r_2+r_3)^2=(r_1+r_2)^2$

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Very impressive and beautiful! Thanks for sharing it! For this problem I have the feeling - every skilled mathematician is capable to solve it and finds something different and interesting. I saw no two people with the same solution.