Example
Here an example of a problem from the United States of America Mathematical Talent Search that can be solved via analytic geometry:
Problem: In a convex pentagon, the sides have lengths, and, though not necessarily in that order. Let, and be the midpoints of the sides, and, respectively. Let be the midpoint of segment, and be the midpoint of segment . The length of segment is an integer. Find all possible values for the length of side .
Solution: Without loss of generality, let, and be located at, and .
Using the midpoint formula, the points, and are located at
- , and
Using the distance formula,
and
Since has to be an integer,
(see modular arithmetic) so .
Read more about this topic: Analytic Geometry
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