Varignon's Theorem states that the figure formed by joining the midpoints of adjacent sides of a convex quadrilateral is always a parallelogram. This parallelogram, known as the Varignon parallelogram, has an area equal to half the area of the original quadrilateral, and a perimeter equal to the sum of the lengths of the original quadrilateral's diagonals.
The Varignon parallelogram is a rhombus (all sides have equal length) if and only if the diagonals of the original quadrilateral have equal length.
The Varignon parallelogram is a rectangle if and only if the diagonals of the original quadrilateral are perpendicular.
Proof of the Theorem
The area of the original quadrilateral is double the magnitude of the cross product of its diagonals. The area of the Varignon parallelogram is the cross product of two of its adjacent sides. The result follows because those sides are parallel to and exactly half the length of the aforementioned diagonals, by the Midline Theorem (an application of similar triangles).
Click on the plot below to add four points which will be the vertices of your convex quadrilateral. Then, click "Add Varignon Parallelogram" to join the midpoints of its sides and create this special parallelogram. Observe the values in the text boxes on the right to verify that the area of the Varignon parallelogram is equal to half the area of the original figure. Select "Show diagonals of original quadrilateral" and compare the perimeter of the Varignon parallelogram to the sum of the lengths of these diagonals.
Area of the Original Quadrilateral:
Area of the Varignon Parallelogram:
Sum of the Lengths of the Diagonals:
Perimeter of the Varignon Parallelogram:
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