Point $G$ is the midpoint of median $\overline{XM}$ of $\triangle XYZ$. Point $H$ is the midpoint of $\overline{XY}$, and point $T$ is the intersection of $\overline{HM}$ and $\overline{YG}$. Find the area of $\triangle MTG$ if $[XYZ] =150$.
I doubt it's actually the AoPS team...Its $\boxed{25/2}$
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To find the area of triangle $\triangle MTG$, let's break it down step by step.
Step 1: Find the area of $\triangle XYZ$.
Given that $[XYZ] = 150$, we know the area of triangle $\triangle XYZ$ is 150.
Step 2: Find the area of $\triangle HYG$.
Since point $H$ is the midpoint of $\overline{XY}$, we can use the concept of similar triangles to find the area of $\triangle HYG$. Triangle $\triangle HYG$ is similar to triangle $\triangle XYZ$, and the ratio of their corresponding sides is $1:2$ (since $H$ is the midpoint of $\overline{XY}$). Therefore, the area of $\triangle HYG$ is $(1/2)^2 = 1/4$ times the area of $\triangle XYZ$. Hence, $[\triangle HYG] = (1/4) \cdot 150 = 37.5$.
Step 3: Find the area of $\triangle MHG$.
Since $G$ is the midpoint of median $\overline{XM}$ of triangle $\triangle XYZ$, we know that $MG = \frac{1}{2}XM$. Since $H$ is the midpoint of $\overline{XY}$, we can also deduce that $\overline{HY} = \frac{1}{2}\overline{XY}$. Therefore, triangle $\triangle MHG$ is similar to triangle $\triangle HYG$ (by the same side ratios).
The ratio of corresponding sides in similar triangles $\triangle MHG$ and $\triangle HYG$ is $1:2$. Thus, the area of $\triangle MHG$ is $(1/2)^2 = 1/4$ times the area of $\triangle HYG$. Hence, $[\triangle MHG] = (1/4) \cdot 37.5 = 9.375$.
Step 4: Find the area of $\triangle MTG$.
Since $\triangle MTG$ is formed by connecting point $T$ (the intersection of $\overline{HM}$ and $\overline{YG}$) to the vertices $M$, $G$, and $T$, we can deduce that triangle $\triangle MTG$ is similar to triangle $\triangle MHG$ (by the same side ratios).
The ratio of corresponding sides in similar triangles $\triangle MTG$ and $\triangle MHG$ is $1:2$. Thus, the area of $\triangle MTG$ is $(1/2)^2 = 1/4$ times the area of $\triangle MHG$. Hence, $[\triangle MTG] = (1/4) \cdot 9.375 = 2.34375$.
Therefore, the area of $\triangle MTG$ is approximately $2.34375$.