Mid-point Theorem and Its Converse [Including Intercept Theorem]
226. The Midpoint Mirror · parallel lines always bisect the third side
Drag E along AC. When DE becomes parallel to BC, it exactly bisects AC.
Converse of the midpoint theorem: a line through the midpoint of one side and parallel to another side bisects the third side. So if D is the midpoint of AB and DE ∥ BC, then E is the midpoint of AC.
What this lesson covers
Try to break it
Drag E along AC. DE only becomes parallel to BC at one position — the exact midpoint of AC, where AE = EC. Try to make DE parallel to BC at any other position; impossible. The converse of the midpoint theorem snaps E to the midpoint.
How you build it
Construct a triangle and a midpoint parallel line.
- Place point A as the apex of the triangle.
- Place point B as the bottom-left vertex.
- Place point C as the bottom-right vertex.
- Draw segment AB.
- Draw segment BC.
- Draw segment CA to complete triangle ABC.
- Midpoint tool: click A then B — D drops at the exact midpoint of AB.
- Using the parallel tool, draw a line through D that is parallel to BC.
- Mark the intersection point E on AC.
The proof, step by step
Prove that a line through the midpoint of one side, parallel to another side, bisects the third side.
- Through C, draw CF parallel to BA, meeting DE produced at F.
- BCFD is a parallelogram (DF // BC and CF // BD)
- CF = BD (Opposite sides of a parallelogram are equal)
- CF = DA (Since BD = DA, given)
- ΔADE ≅ ΔCFE (ASA: AD=CF, ∠DAE=∠ECF, ∠ADE=∠EFC)
- AE = EC (Corresponding parts of congruent triangles are equal)
Worked example
In ΔABC, D is the midpoint of side AB. A line through D parallel to BC meets AC at E. If AC = 16 cm, what is the length of AE?
By the Converse of the Mid-point Theorem, since D is the midpoint of AB and DE || BC, E must be the midpoint of AC. Therefore, AE = AC / 2 = 16 / 2 = 8 cm.
- 6 cm
- 7 cm
- 8 cm — correct
- 9 cm