Integer triangles sharing angles

God plays dice

Here’s another brute-force number theory puzzle, in the spirit of the Russian puzzle I looked at a couple weeks ago. From what would Martin Gardner tweet: “Two triangles share one angle but have six different side lengths (all whole numbers between 1 and 9). Find those lengths.”

Do we expect such triangles to exist? Well, clearly there are lots of different-sized right triangles, which share an angle. To take the smallest possible example, 3-4-5 and 6-8-10 are both the sides of right triangles, and they share an angle. But sadly, 10 is too large…

This is a bit tricky to think about, because the angles of an integer-sided triangle aren’t going to be anything nice. So instead of looking at the angles themselves let’s look at their cosines. Given a triangle with sides having lengths (a, b, c), the angle C, opposite the side of length c satisfies

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