Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.7 AIME Prove it United States

Problem:

Let $M$ be the midpoint of the side $AC$ of triangle $ABC$ . If $N$ is the point on the side $AB$ , $O$ intersection of the lines $BM$ and $CN$ , and if the areas of triangles $BON$ and $COM$ are equal, prove that $N$ is the midpoint of $AB$ .

Solution

Solution:

Since the areas of $\triangle BON$ and $\triangle COM$ are equal we see that the areas of triangles $\triangle BCN$ and $\triangle CBM$ are also equal. Since these two triangles share the side, they must have the corresponding altitudes equal. Hence the length of perpendiculars from $M$ and $N$ to $BC$ are equal, implying that $NM \parallel BC$ . Thus $MN$ is the midsegment of $\triangle ABC$ and consequently $N$ is the midpoint of $AB$ .

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