Here is the question:
$A$, $B$, $C$ are three points on a horizontal line through the base $O$ of a pillar $OP$, such that $OA$, $OB$, $OC$ are in $AP$. If $\alpha$, $\beta$, $\gamma$ the angles elevation of the top of the pillar at $A$, $B$, $C$ respectively are also in A.P, then $\sin \alpha$, $\sin \beta$ ,$\sin \gamma$ are in AP, GP or HP?
Here AP stands for arithmetic progression, GP for geomtric progression and HP for harmonic progression.
Here is my process which is fairly straightforward: $$ \begin{align*} OA+OC &=2\cdot OB \\ \implies OP \cot \alpha + OP \cot \gamma &= 2 \cdot OP \cot \beta \\ \implies \dfrac{\cos \alpha}{\sin \alpha}+\dfrac{\cos \gamma }{\sin \gamma} & = 2 \cdot \dfrac{\cos \beta}{\sin \beta} \\ \implies \cos \alpha \cdot \sin \gamma + \cos \gamma \cdot \sin \alpha & = 2 \cdot \dfrac{\cos \beta \cdot \sin \alpha \cdot \sin \gamma}{\sin \beta} \\ \implies \sin (\alpha + \gamma) = \sin (2 \beta) & = 2 \cdot \dfrac{\cos \beta \cdot \sin \alpha \cdot \sin \gamma}{\sin \beta} \\ \implies 2 \cdot \sin \beta \cos \beta & = 2 \cdot \dfrac{\cos \beta \cdot \sin \alpha \cdot \sin \gamma}{\sin \beta} \\ \implies \sin^2 \beta & = \sin \alpha \cdot \sin \gamma \end{align*} $$
Which proves that they are in GP.
However, here is my problem:
If I find the angles $APB$ and $BPC$ I notice that they are same ($\alpha - \beta = \beta - \gamma$). Which means $PB$ is angle bisector. After applying angle bisector theorem we get that $AP=PC$ but that can only happen if $A$,$B$, and $C$ coincide. Which tells that $\alpha=\beta=\gamma$ and that $\sin \alpha= \sin \beta= \sin \gamma$ and hence they are in AP, GP as well as HP.
Here is another approach I tried: Let $OA=a$, $OB=a+d$ and $OC=a+2d$.
$$\tan 2 \beta= \tan( \alpha + \gamma)$$
where $\tan \alpha = OP/OA$, $\tan \beta = OP/OB$ and $\tan \gamma = OP/OC$ Putting the values of the above expressions in the form of $a$, $d$ and $OP$ and then simplifying we get $d=0$ which further confirms that $OA=OB=OC$.
Am I going wrong anywhere?
Edit: Sorry if I confused anyone, but angle of elevation can be thought of as the angle the pillar $OP$ subtends at the point $A$. So $\angle OAP = \alpha$.