Midpoint theorem (triangle)

Last updated October 24, 2024

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]

Proof

Proof by construction

Proof

Given: In a $\triangle ABC$ 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:

$MN\parallel BC$
$MN={1 \over 2}BC$

Proof:

$AN=CN$ (given)
$\angle ANM=\angle CND$ (vertically opposite angle)
$MN=DN$ (constructible)

Hence by Side angle side.

\triangle AMN\cong \triangle CDN

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

$AM=BM=CD$
$\angle MAN=\angle DCN$

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

$AM\parallel CD\parallel BM$

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

$MN\parallel BC$
$MN={1 \over 2}MD={1 \over 2}BC$ ,

Q.E.D.

Proof by similar triangles

Proof

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

To prove:

$DE\parallel AB$ ,
$DE={\frac {1}{2}}AB$ .

Proof:

$\angle C$ is the common angle of $\triangle ABC$ and $\triangle DEC$ .

Since DE connects the midpoints of AC and BC, $AD=DC$ , $BE=EC$ and ${\frac {AC}{DC}}={\frac {BC}{EC}}=2.$ As such, $\triangle ABC$ and $\triangle DEC$ are similar by the SAS criterion.

Therefore, ${\frac {AB}{DE}}={\frac {AC}{DC}}={\frac {BC}{EC}}=2,$ which means that $DE={\frac {1}{2}}AB.$

Since $\triangle ABC$ and $\triangle DEC$ are similar and $\triangle DEC\in \triangle ABC$ , $\angle CDE=\angle CAB$ , which means that $AB\parallel DE$ .

Q.E.D.

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References

↑ 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.
↑ 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.

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

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[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.

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