# 1971 Canadian MO Problems/Problem 9

## 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