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Problem 1069

AMC 12 late, AIME early

Geometry Difficulty 4.7 Prove it Berkeley Math Circle Monthly Contest 4 · United States

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 $AB$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official 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 $AB$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.