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Problem

Two flag poles of height $h$ and $k$ are situated $2a$ units apart on a level surface. Find the set of all points on the surface which are so situated that the angles of elevation of the tops of the poles are equal.

Solution

Let the top of the flagpole of height h be denoted by H and its intersection with the surface be denoted by I. Similarly, let the top of the flagpole of length k be denoted by K and its intersection with the surface be denoted by J. Finally, let the desired point be P. Reflect H over IJ to H'. Since angles are preserved in reflections, we have < HPI = < H'PI. However, the problem statement tells us that < HPI = < KPJ. By transitivity, we must have < H'PI = < KPJ, implying that H', P, and K are collinear by the converse of Vertical Angles. In effect, we can construct our desired point P by reflecting H' over IJ and finding the intersection of H'K and IJ. Therefore, there is only one such point that is constructed with the method shown above.

-Solution by thecmd999