Triangle Radicals

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An integer-sided triangle always has an area of the form , where is a square-free number and is an integer. Denote , which appears under the radical sign, as the "radical of the triangle".


This Demonstration gives up to 200 triangles for each of the square-free number under 1000. A Heronian triangle (having rational sides and rational area) has radical 1.


Contributed by: Ed Pegg Jr (May 2016)
Open content licensed under CC BY-NC-SA



For all triangles with a given square-free radical , the value is also divisible by the term of , where (mod12).

In the Demonstration "Wheel Graphs with Integer Edges", it was shown that all small integer-edge wheel graphs use triangles of two or fewer radical types.

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