1974 IMO Problems/Problem 2
In the triangle ABC; prove that there is a point D on side AB such that CD
is the geometric mean of AD and DB if and only if
.
Solution
Since this is an "if and only if" statement, we will prove it in two parts.
Before we begin, note a few basic but important facts.
1. When two variables and
are restricted by an equation
for some constant
, the maximum of their product occurs when
.
2. The triangle inequality states that for a triangle with sides ,
, and
fulfills
, meaning that
or
, which is equivalent to saying that
and
both cannot be less than or equal to
.
3. As point D moves along the base of the side AB, the locus of points where C can exist fills in a semicircle because C must be a distance
Part 1: If , then such a point D exists.
First note that in order for D to exist, the line segment CD must be able to reach from the vertex C to the side AB, meaning that