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May 31st, 2014, 09:35 AM   #1
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Rational ratios in triangle

Determine all triangles ABC such that all of AB/BC,BC/CA,CA/AB,∠A/∠B,∠B/∠C,∠C/∠A are rational
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May 31st, 2014, 01:15 PM   #2
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There are infinite number of such triangles.
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May 31st, 2014, 05:21 PM   #3
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Why? How it can be justified?
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May 31st, 2014, 05:58 PM   #4
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Let edges be a, b, c, and angles be $\displaystyle \alpha , \; \beta , \; \gamma$.

$\displaystyle a, \; b, \; c, \; \alpha , \; \beta , \; \gamma \in \mathbb{R}$ and $\displaystyle a/b, \; b/c, \; c/a, \; \alpha / \beta , \; \beta / \gamma , \; \gamma / \alpha \in \mathbb{Q}$

Example is equilateral triangle.

It is not easy to solve, hence edges and angles are not necessarily rational. Is there any other hints?
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May 31st, 2014, 07:05 PM   #5
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There is not any hint here... It's really difficult
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May 31st, 2014, 10:14 PM   #6
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tahirimanov is correct for a literal reading of the question, for the reason he stated. But the question remains unanswered for the equivalence class of triangles up to similarity.
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