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Midpoint theorem (triangle)

From Wikipedia, the free encyclopedia

The midpoint 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.

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

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Proof

Given: In a the points M and N are the midpoints of the sides AB and AC respectively.

Construction: MN is extended to D where MN=DN, join C to D.

To Prove:

Proof:

  • (given)
  • (vertically opposite angle)
  • (constructible)

Hence by Side angle side.

Therefore, the corresponding sides and angles of congruent triangles are equal

Transversal AC intersects the lines AB and CD and alternate angles ∠MAN and ∠DCN are equal. Therefore

Hence BCDM is a parallelogram. BC and DM are also equal and parallel.

References

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  • French, Doug (2004). Teaching and Learning Geometry. Bloomsbury. pp. 81–84. ISBN 9780826473622. (online copy, p. 81, at Google Books)
  • Clapham, Christopher; Nicholson, James (2009). The Concise Oxford Dictionary of Mathematics. Oxford University Press. p. 297. ISBN 9780199235940. (online copy, p. 297, at Google Books)
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