Midpoint theorem (triangle)

Given: In a ${\displaystyle \triangle ABC}$ 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:

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

Proof:

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

Hence by Side angle side.

${\displaystyle \triangle AMN\cong \triangle CDN}$

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

• ${\displaystyle AM=BM=CD}$
• ${\displaystyle \angle MAN=\angle DCN}$

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

• ${\displaystyle AM\parallel CD\parallel BM}$

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

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

Q.E.D.

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

To prove:

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

Proof:

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

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

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

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

Q.E.D.