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:

\(\alpha\), \(\beta\), \(\gamma\) 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}$$

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.*