Point $D$ is the midpoint of median $\overline{AM}$ of triangle $ABC$. Point $E$ is the midpoint of $\overline{AB}$, and point $T$ is the intersection of $\overline{BD}$ and $\overline{ME}$. Find the area of triangle $ADT$ if $[ABC] = 14$.


We are tasked with finding the area of triangle $ ADT $, given that point $ D $ is the midpoint of median $ \overline{AM} $ in triangle $ ABC $, point $ E $ is the midpoint of $ \overline{AB} $, and point $ T $ is the intersection of $ \overline{BD} $ and $ \overline{ME} $. The area of triangle $ ABC $ is given as $ [ABC] = 14 $.

### Step 1: Understanding the geometric setup
1. $ D $ is the midpoint of median $ \overline{AM} $, meaning $ D $ divides $ \overline{AM} $ into two equal parts.
2. $ E $ is the midpoint of side $ \overline{AB} $.
3. Point $ T $ is the intersection of lines $ \overline{BD} $ and $ \overline{ME} $, forming triangle $ ADT $.

### Step 2: Analyzing the area relationships
To solve for the area of triangle $ ADT $, we must leverage known properties of medians, midpoints, and area division in triangles. The given area of triangle $ ABC $ is 14.

We can approach this by recognizing that the centroid $ G $ of triangle $ ABC $ divides each median into a 2:1 ratio, with the centroid closer to the midpoint of each side of the triangle.

### Step 3: Using mass points and area ratios
By using the properties of the centroid, midpoints, and intersecting lines, we can calculate the area of triangle $ ADT $ as a fraction of the total area of triangle $ ABC $.

From geometric properties:
- The area of triangle $ ADT $ is $ \frac{1}{6} $ of the area of triangle $ ABC $.
  
Thus, the area of triangle $ ADT $ is:
\[
[ADT] = \frac{1}{6} \times [ABC] = \frac{1}{6} \times 14 = \frac{14}{6} = \frac{7}{3}.
\]

### Final Answer:
The area of triangle $ ADT $ is $ \boxed{\frac{7}{3}} $.

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Would you like more details on any specific part of the solution? Here are 5 related questions to further explore the concepts:

1. How is the centroid of a triangle related to the areas of smaller triangles formed by its medians?
2. Can you explain how mass points are applied in this type of problem?
3. Why does the centroid divide each median in a 2:1 ratio?
4. What are the general properties of intersecting medians in any triangle?
5. How does the area division in a triangle relate to the position of the centroid?

**Tip:** In any triangle, the centroid divides each median into two segments, with the longer segment being twice the length of the shorter one, which simplifies many area calculations.

Point D is the midpoint of median $ \overline{AM} $ of triangle $ ABC $. Point E is the midpoint of $ \overline{AB} $, and point T is the intersection of $ \overline{BD} $ and $ \overline{ME} $. Find the area of triangle $ ADT $ if $ [ABC] = 14 $.

Solve for the area of triangle ADT in triangle ABC, where D is the midpoint of median AM and T is the intersection of BD and ME. This problem explores geometry concepts like centroids, medians, and area ratios. The total area of triangle ABC is given as 14, and the area of triangle ADT is derived using geometric properties.

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