Saturday, February 4, 2017

sequences and series - Sides of a triangle in arithmetic progression. Find possible values of common difference



Question



If the sides of a triangle are in arithmetic progression with first term $1$ and

common difference $d$ find the set of possible values of $d$



Attempt



I do not have much of a clue on how to go about solving this. I've tried letting the angles be $\theta,$ $\theta+\varepsilon$ and $\pi-2\theta-\varepsilon$ and using the sine rule, and even the cosine rule, but to no prevail.



Any hints would be very much appreciated!


Answer



Thank you dxiv and Zubin. I had forgotten all about them.




Using the triangle inequality (multiple times) we get $$-\frac{1}{3}< d< 1.$$


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