# Triangle Inequality

1. Oct 10, 2012

### sharpycasio

1. The problem statement, all variables and given/known data
Prove the following inequality for any triangle that has sides a, b, and c.

$$-1<\frac{a}{b}+\frac{b}{c}+\frac{c}{a}-\frac{b}{a}-\frac{a}{c}-\frac{c}{b}<1$$

2. Relevant equations
3. The attempt at a solution

I think we have to use sine or cosine at a certain point because the bounds of the inequality are the same as the bounds of the two functions' ranges. Perhaps the Sine Law since that applies to all triangles? Tried rearranging it, pairing up the reciprocals. Maybe the fractions represent ratios ($sin(\theta)$)

$$-1<(\frac{a}{b}-\frac{b}{a})+(\frac{b}{c}-\frac{c}{b})+(\frac{c}{a}-\frac{a}{c})<1$$

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Oct 10, 2012

### LCKurtz

3. Oct 11, 2012

### sharpycasio

I am sorry for reposting the same question. It's just that I've been working on it for hours and I really have to solve it for tomorrow. My apologies.