Triangle Mid-Segment Theorem

Take any triangle ABC. Find the mid-point D of AB and the mid-point E or AC.

Click here to open a GSP file. Explore with measures of DE and BC as any vertex is moved.

One of the most useful theorems in geometry of plane triangles is the Triangle Mid-Segment Theorem:

In any triangle, a segment joining the midpoints of any two sides will be parallel to the third side and half its length.

Prove the Triangle Mid-Segment Theorem.

If the figure above, we are given that D is the mid-point of AB and E is the mid-point of AC. So we know that AD = DB and AE = EC.

In such situations it is often useful to draw some auxiliary figures and try to find congruent or similar triangles.

Hint. An auxiliary figure.

Extension. State and extend or adapt the theorem to trapezoids and parallelograms.

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