Quadrilateral is Parallelogram iff Diagonals Bisect each other
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Theorem
Let $ABCD$ be a quadrilateral.
Then:
- $ABCD$ is a parallelogram
Proof
Sufficient Condition
Let $ABCD$ be a parallelogram.
Then by Diameters of Parallelogram Bisect each other:
- $AD$ is a bisector of $BC$
and
- $BC$ is a bisector of $AD$.
$\Box$
Necessary Condition
Let $ABCD$ be such that:
- $AD$ is a bisector of $BC$
and
- $BC$ is a bisector of $AD$.
Then from Quadrilateral with Bisecting Diagonals is Parallelogram, $ABCD$ is a parallelogram.
$\blacksquare$
Sources
- 1968: M.N. Aref and William Wernick: Problems & Solutions in Euclidean Geometry ... (previous) ... (next): Chapter $1$: Triangles and Polygons: Theorems and Corollaries $1.23 \ \text{(iv)}$