The midpoint theorem
Join two midpoints of a triangle, and something exact appears.
- Triangle ABC. D is the midpoint of AB. E is the midpoint of AC.
- Segment DE joins the two midpoints.
- DE runs parallel to BC.
- BC measures 10. DE measures exactly 5.
The rule
Midpoint theorem: Join two midpoints of a triangle. The segment is parallel to the third side. It is exactly half that side.
A midpoint is the point halfway along a side. A segment is the part of a line between two points.
Why? Extend DE past E to a point F, so EF equals DE. Triangle ADE lands on triangle CFE by SAS. "Lands on" means fits exactly on top. SAS: two sides and the angle between them match. AE equals EC and DE equals FE. The angles at E are vertically opposite.
Matching parts give CF equal to AD. Angles DAE and FCE match. They are an equal alternate pair. So CF is parallel to AD. D is the midpoint of AB, so AD equals DB. So CF is equal and parallel to DB too. So DBCF is a parallelogram. DF equals BC, and DE is half of DF. So DE is half of BC. DF is also parallel to BC. So DE is parallel to BC.
| Side | Length |
|---|---|
| BC | 10 |
| DE | 5 |
Use it both ways. If DE is 5.5, then BC is 11. If BC is 11, then DE is 5.5.
The reverse is true too. Let D be the midpoint of AB. A line through D, parallel to BC, bisects AC.
Move the triangle. Watch DE stay half of BC.
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