Midpoint theorem (triangle)

Last updated • 2 min readFrom Wikipedia, The Free Encyclopedia
D and E midpoints of AC and BC
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{\displaystyle {\begin{aligned}&{\text{D and E midpoints of AC and BC}}\\\Rightarrow \,&DE\parallel AB{\text{ and }}2|DE|=|AB|\end{aligned}}} Midpoint theorem.svg

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]

Contents

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 construction

Proof
Midpoint Theorem proof.png

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.

  • ,

Q.E.D.

Proof by similar triangles

Proof
Midpoint theorem.svg

Let D and E be the midpoints of AC and BC.

To prove:

  • ,
  • .

Proof:

is the common angle of and .

Since DE connects the midpoints of AC and BC, , and As such, and are similar by the SAS criterion.

Therefore, which means that

Since and are similar and , , which means that .

Q.E.D.

See also

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References

  1. Clapham, Christopher; Nicholson, James (2009). The concise Oxford dictionary of mathematics: clear definitions of even the most complex mathematical terms and concepts. Oxford paperback reference (4th ed.). Oxford: Oxford Univ. Press. p. 297. ISBN   978-0-19-923594-0.
  2. French, Doug (2004). Teaching and learning geometry: issues and methods in mathematical education. London; New York: Continuum. pp. 81–84. ISBN   978-0-8264-7362-2. OCLC   56658329.