Diagonals that bisect each other and parallelogram

Another sufficient condition for the parallelogram concerns the diagonals: if they cut each other in half, the quadrilateral is a parallelogram. The proof uses vertically opposite angles and the first criterion.

Theorem — Diagonals that bisect each other

If a quadrilateral has diagonals that bisect each other, then it is a parallelogram.

Proof

We show that the pairs of triangles formed by the diagonals are congruent, and from this we derive the parallelism.

1. Hypothesis. The diagonals $AC$ and $BD$ intersect at $M$ with $AM\cong MC$ and $BM\cong MD$.
2. Observe. $\widehat{AMB}\cong\widehat{CMD}$ (vertically opposite angles).
3. Deduce. By the first criterion: $\triangle AMB\cong\triangle CMD$, hence $AB\cong CD$ and $\widehat{ABM}\cong\widehat{CDM}$ (alternate interior) $\implies AB\parallel CD$.
4. Similarly. $\widehat{AMD}\cong\widehat{CMB}$ and $\triangle AMD\cong\triangle CMB \implies AD\parallel BC$.
5. Verified. Hence $ABCD$ is a parallelogram.

$\blacksquare$

Links

Topics: Euclidean geometry
Concepts: Alternate interior angles · Vertically opposite angles · Congruence criteria · Proof · Parallelogram
Skills: Proving · Synthetic geometry