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Macedonian Mathematical Competitions

Geometry Difficulty 4.5 AIME Prove it North Macedonia

Let $p$ be a given line. $A$ and $B$ are two points such that $A \notin p$ and $B \notin p$ . Construct a point $C$ on the line $p$ such that $\overline{AC} = \overline{BC}$ .

Solution

The points that lie on the bisector of the segment $AB$ are on equal distance of its end points $A$ and $B$ . Hence the desired point $C$ is the intersection of the bisector of $AB$ and the line $p$ . The problem has a unique solution if $AB$ and $p$ are not perpendicular, if $AB$ and $p$ are perpendicular and $p$ is not the bisector of $AB$ then the problem doesn't have a solution and if $AB$ and $p$ are perpendicular and $p$ is the bisector of $AB$ then the problem has infinitely many solution because every point of $p$ is a solution.

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