## Angles and Tangents in Arithmetic Progression

For First-Quadrant angles $$\alpha$$ and $$\gamma\;$$ (and $$\beta$$ and $$\beta^\prime$$), such that
$$\alpha$$, $$\beta$$, $$\gamma$$ and $$\tan\alpha$$, $$\tan\beta^\prime$$, $$\tan\gamma$$ are both arithmetic progressions:
\large\begin{align} 2\,\beta\; &\;=\; \alpha + \gamma \\[4pt] 2\tan\beta^\prime &\;=\; \tan\alpha + \tan\gamma \end{align}\quad\Longrightarrow\quad \begin{array}{c} \beta \;\leq\; \beta^\prime \\ \text{with equality when and only when} \\ \alpha = \beta = \beta^\prime = \gamma \end{array}

Motivated by this question on the Mathematics Stack Exchange.