All the equations in geometry between segments are related to area or line or are nondimensional relations related to trigonometry.Is there a relation in geometry in which the third power of a segment is a function of other segments of a triangle?
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Maybe someone can build off of this idea, but considering cube roots are impossible within compass and straightedge math (having a lot to do with circles and triangles), perhaps not. – Graviton Aug 11 '20 at 07:21
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Assuming that the "segments" of a triangle are the sides: If I'm understanding the question you're asking whether there exists a function $f$ such that if $A,B,C$ are the sides of a triangle then $$A^3=f(B,C).$$
The answer is obviously no. Has nothing to do with constructibility, as suggested in a comment. Consider a triangle with sides $A,B,C$ and another triangle with sides $A',B,C$, with $A'\ne A$.
Arctic Char
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David C. Ullrich
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