While designing another problem I came up with the following question:

Considering the picture below, can you determine $h$ based only on the information of $\alpha$, $\beta$ and $x$ given the fact that the segment $x$ is the continuation of the height?

My guess is that the problem is not solvable (in the sense that one can construct multiple $h$ with the same $\alpha,\beta$ and $x$), but cannot see how to show this.

So far I gave the problem a try by splitting the angle and combining some trigonometric identities but couldn't conclude.

So far I only obtained the following \begin{align*} \tan(\alpha)=\frac{(h_1+h_2)(x+y)}{(x+y)^2-h_1h_2} \end{align*} and $$\tan(\beta)=\frac{(h_1+h_2)\cdot y}{y^2-h_1h_2}.$$

Many thanks in advance.

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