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Referring to the diagram above, [[triangle]]s ''ADC'' and ''HDG'' are similar by the side-angle-side criterion, so [[angle]]s ''DAC'' and ''DHG'' are equal, making ''HG'' parallel to ''AC''. In the same way ''EF'' is parallel to ''AC'', so ''HG'' and ''EF'' are parallel to each other; the same holds for ''HE'' and ''GF''.
Varignon's theorem can also be proved as a theorem of [[affine geometry]] organized as [[linear algebra]] with the linear combinations restricted to coefficients summing to 1, also called affine or [[Barycentric coordinates (mathematics)|barycentric coordinates]]. The proof applies even to skew quadrilaterals in spaces of any dimension.
Any three points ''E'', ''F'', ''G'' are completed to a parallelogram (lying in the plane containing ''E'', ''F'', and ''G'') by taking its fourth vertex to be ''E'' − ''F'' + ''G''. In the construction of the Varignon parallelogram this is the point (''A'' + ''B'')/2 − (''B'' + ''C'')/2 + (''C'' + ''D'')/2 = (''A'' + ''D'')/2. But this is the point ''H'' in the figure, whence ''EFGH'' forms a parallelogram.
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