Midpoint theorem (triangle) explained

The midpoint theorem, midsegment theorem, or midline theorem states that if the midpoints of two sides of a triangle are connected, then the resulting line segment will be parallel to the third side and have half of its length. The midpoint theorem generalizes to the intercept theorem, where rather than using midpoints, both sides are partitioned in the same ratio.^{[1] [2]}

The converse of the theorem is true as well. That is if a line is drawn through the midpoint of triangle side parallel to another triangle side then the line will bisect the third side of the triangle.

The triangle formed by the three parallel lines through the three midpoints of sides of a triangle is called its medial triangle.

Proof

Proof by similar triangles

See also

• Median of the Trapezoid theorem

External links

• The midpoint theorem and its converse
• The Mid-Point Theorem
• Midpoint theorem and converse Euclidean explained Grade 10+12 (video, 5:28 mins)
• midpoint theorem at the Proof Wiki

Notes and References

1. Book: Clapham, Christopher . The concise Oxford dictionary of mathematics: clear definitions of even the most complex mathematical terms and concepts . Nicholson . James . 2009 . Oxford Univ. Press . 978-0-19-923594-0 . 4th . Oxford paperback reference . Oxford . 297.
2. Book: French, Doug . Teaching and learning geometry: issues and methods in mathematical education . 2004 . Continuum . 978-0-8264-7362-2 . London; New York . 81–84 . ocm56658329.