GeoProof — Parallelogram Diagonals Bisect Each Other

A parallelogram ABCD is shown. You can drag corners A, B, and D; corner C moves automatically to keep the shape a parallelogram. Both diagonals are drawn: AC and BD. They cross at a point M. The theorem: M is the midpoint of each diagonal, so the diagonals bisect each other — AM equals MC and BM equals MD. Equal-length tick marks show the matching halves, and the four half-lengths are shown live and stay paired. An info button opens a drawer with the explanation.

AM = MC and BM = MD

Diagonals Bisect Each Other

In every parallelogram, the two diagonals cross at their common midpoint:

AM = MC and BM = MD

Each diagonal cuts the other exactly in half, so their intersection M is the midpoint of both.

Why it's true

Opposite sides AB and DC are equal and parallel, which makes triangles ABM and CDM congruent (ASA). Matching sides then give AM = CM and BM = DM.

It works in reverse

The converse holds too: if a quadrilateral's diagonals bisect each other, it must be a parallelogram — a handy test, and the reason M is the figure's centre of symmetry.

Special cases

In a rectangle the diagonals are also equal; in a rhombus they're also perpendicular; in a square, both at once.