1974 IMO Problems/Problem 2
In the triangle , prove that there is a point on side such that is the geometric mean of and if and only if .
Let a point on the side . Let the altitude of the triangle , and the symmetric point of through . We bring a parallel line from to . This line intersects the ray at the point , and we know that .
The distance between the parallel lines and is .
Let the circumscribed circle of , and the perpendicular diameter to , such that are on difererent sides of the line .
In fact, the problem asks when the line intersects the circumcircle. Indeed:
Suppose that is the geometric mean of .
Then, from the power of we can see that is also a point of the circle . Or else, the line intersects
where is the altitude of the isosceles .
We use the formulas:
So we have
Then we can go inversely and we find that the line intersects the circle (without loss of generality; if then is tangent to at )
So, if then for the point we have and
The above solution was posted and copyrighted by pontios. The original thread for this problem can be found here: 
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