Angles and Tangents in Arithmetic Progression

trigonograph-anglesandtangentsinarithmeticprogression

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.