May 31st, 2014, 09:35 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, 01:15 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, 05:21 PM  #3 
Newbie Joined: May 2014 From: Washington Posts: 3 Thanks: 0 
Why? How it can be justified?

May 31st, 2014, 05:58 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, 07:05 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, 10:14 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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