Finding Cartesian coordinates of remaining vertices of triangle, given a vertex and an angle

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I have an isosceles triangle $ABC$, where the height $h$ and the angle at vertex $A$ are known. The Cartesian coordinates of vertex $A$ are also known to be $\left(x,y\right)$. is it possible to find the coordinates of the remaining vertices?

Triangle ABC

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It is not possible to find the coordinates of the remaining two vertices.

Rotate the triangle about A. The height remains the same. Angle A remains the same. The coordinates of A remain the same but the coordinates of B and C change.

Therefore, there is not enough information to find the coordinates of the remaining vertices.