If you solve the system of quadratic equations in six variables, it looks like a positive answer would be miraculous. To try and get a positive result, I changed the question slightly: Is there a set of three right triangles with hypotenuse Hn and area An such that H1+H2=H3 and A1+A2=A3? The system of equations is still a mess, but I did hit a TRUE, for Pythagorean triples: {6, 8, 10}, {33, 44, 55}, {25, 60, 65}. and pasted a depiction here: https://0x0.st/ia19.png

From the image it's clear that the two smaller triangles are similar (the scale factor is 11/2), which certainly makes a hit more likely, but I don't know if it's a necessary condition.

Are there any more triples of triples TRUE for this SAT? If yes, does it turn out again that the two smaller triangles are similar? I looked a bit higher but didn't find anything else. Cheers, Brad On Sat, Mar 21, 2020 at 8:08 AM Richard <rihess@cox.net> wrote:

In these trying times here is a problem for which I know of no solution. Do there exist three Pythagorean triangles which have the same hypotenuse such that the area of one is the sum of the areas of the other two? Happy puzzling, Dick Hess

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