I'll throw out a dumb idea: can anyone find a rational point on
UPDATE: The above formula used to have a sign error, which I have just fixed, and Bjorn's reponse was to the version with the sign error. Thanks to Kevin Buzzard for pointing this out to me.
Because, if so, gives triangles with rational area. Of course, this still wouldn't give an integer solution, but it would rule out a number of easy arguments against one existing.
I did a brute force search of values of with numerator and denominator under 5000 and didn't find any, but I don't think that is large enough to even count as evidence against one existing.
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