To solve the problem, we need to analyze the situation where the line segments joining the midpoints of the sides of a triangle form four smaller triangles. We will demonstrate that each of these smaller triangles is similar to the original triangle.
### Step-by-Step Solution:
1. **Identify the Triangle and Its Midpoints:**
Let triangle ABC be given, where A, B, and C are the vertices. We will find the midpoints of the sides AB, BC, and AC. Let D, E, and F be the midpoints of sides AB, BC, and AC respectively.
2. **Draw the Midsegments:**
Connect points D and F, and points E and D. This creates new triangles: triangle ADF, triangle BDE, triangle CEF, and triangle DEF.
3. **Establish Parallel Lines:**
By the properties of midsegments in a triangle, we know that line segment DF is parallel to side BC, and line segment DE is parallel to side AC. This is due to the Midsegment Theorem, which states that a segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.
4. **Use Corresponding Angles:**
Since DF is parallel to BC, the angles ∠ADF and ∠ABC are equal (corresponding angles). Similarly, since DE is parallel to AC, the angles ∠ADE and ∠CAB are equal.
5. **Show Similarity of Triangles:**
Now we can show that triangle ADF is similar to triangle ABC by the Angle-Angle (AA) similarity criterion:
- ∠ADF = ∠ABC (corresponding angles)
- ∠ADE = ∠CAB (corresponding angles)
Therefore, triangle ADF ~ triangle ABC.
6. **Repeat for Other Triangles:**
Similarly, we can show that:
- Triangle BDE is similar to triangle ABC.
- Triangle CEF is similar to triangle ABC.
- Triangle DEF is also similar to triangle ABC.
7. **Conclusion:**
Thus, we conclude that each of the four triangles (ADF, BDE, CEF, and DEF) formed by the midpoints of triangle ABC is similar to the original triangle ABC.
