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 May 31st, 2014, 08:35 AM #1 Newbie   Joined: May 2014 From: Washington Posts: 3 Thanks: 0 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
 May 31st, 2014, 12:15 PM #2 Senior Member     Joined: Nov 2013 From: Baku Posts: 502 Thanks: 56 Math Focus: Geometry There are infinite number of such triangles.
 May 31st, 2014, 04:21 PM #3 Newbie   Joined: May 2014 From: Washington Posts: 3 Thanks: 0 Why? How it can be justified?
 May 31st, 2014, 04:58 PM #4 Senior Member     Joined: Nov 2013 From: Baku Posts: 502 Thanks: 56 Math Focus: Geometry 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?
 May 31st, 2014, 06:05 PM #5 Newbie   Joined: May 2014 From: Washington Posts: 3 Thanks: 0 There is not any hint here... It's really difficult
 May 31st, 2014, 09:14 PM #6 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 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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