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Convergence May 31st, 2014 08:35 AM

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

tahirimanov May 31st, 2014 12:15 PM

There are infinite number of such triangles.

Convergence May 31st, 2014 04:21 PM

Why? How it can be justified?

tahirimanov May 31st, 2014 04:58 PM

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?

Convergence May 31st, 2014 06:05 PM

There is not any hint here... It's really difficult

CRGreathouse May 31st, 2014 09:14 PM

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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