1. The problem is to understand the Midpoint Theorem and its converse. 2. The Midpoint Theorem states: If a line segment joins the midpoints of two sides of a triangle, then this segment is parallel to the third side and half as long. 3. Mathematically, if $M$ and $N$ are midpoints of sides $AB$ and $AC$ of triangle $ABC$, then: $$MN \parallel BC \quad \text{and} \quad MN = \frac{1}{2} BC$$ 4. The converse of the Midpoint Theorem states: If a line segment joining a point on one side of a triangle is parallel to the third side and half its length, then the point is the midpoint of that side. 5. To prove the Midpoint Theorem, consider triangle $ABC$ with $M$ and $N$ midpoints of $AB$ and $AC$ respectively. 6. By joining $M$ and $N$, we form segment $MN$. 7. Using coordinate geometry or vector methods, it can be shown that $MN$ is parallel to $BC$ and $MN = \frac{1}{2} BC$. 8. The converse uses the same logic in reverse: if $MN$ is parallel to $BC$ and $MN = \frac{1}{2} BC$, then $M$ and $N$ must be midpoints. This theorem is useful in geometry for proving properties of triangles and solving related problems.