The Midpoint Theorem

The Midpoint Theorem is a fundamental result in Euclidean geometry that establishes a relationship between the midpoints of two sides of a triangle and the third side. It is a specific case of the more general Intercept Theorem (or Thales' theorem) and is essential for solving problems involving parallel lines, midpoints, and proportional segments in triangles. Definition of a Midpoint: A midpoint of a line segment is the point that divides the segment into two equal parts. If M is the midpoint of AB, then AM = MB. The Midpoint Theorem Statement: In a triangle, the line segment joining the midpoints of two sides is parallel to the third side and is equal to half the length of the third side. Formal Statement: Consider triangle ABC. Let D be the midpoint of side AB, and let E be the midpoint of side AC. Then: