2006 Indonesia MO Problems/Problem 5
In triangle , is the midpoint of side and is the centroid of triangle . A line passes through , intersecting line at and line at , where and . If denotes the area of triangle , show that .
Solution (credit to Moonmathpi496)
First, since is the centroid of the triangle and , . Also, note that , so . Similarly, . Substituting the areas results in Draw such that , passes through , is on , and and . By AA Similarity, and . Since , lengths on and are the lengths of and , respectively. By applying Menelaus' Theorem, . Note that , so and .
Adding to both sides results in , and rearranging both sides results in . Adding to both sides results in .
Note that and , so substituting those values results in . Thus, , so .
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