Varignon Parallelogram of a Quadrilateral — Definition

Varignon Parallelogram of a Quadrilateral

The parallelogram formed by connecting the midpoints of adjacent sides of a quadrilateral.

A quadrilateral with its four midpoints connected, forming an inner parallelogram (Varignon Parallelogram) outlined within.

Key Formula

M = \left(\frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2}\right)

\text{Area of Varignon parallelogram} = \frac{1}{2} \times \text{Area of original quadrilateral}

Where:

$M$ = Midpoint of a side of the quadrilateral
$(x_1, y_1),\;(x_2, y_2)$ = Coordinates of the two endpoints of that side
$\tfrac{1}{2}$ = The Varignon parallelogram always has exactly half the area of the original quadrilateral

Worked Example

Problem: A quadrilateral has vertices A(0, 0), B(6, 0), C(8, 4), and D(2, 6). Find the vertices of its Varignon parallelogram and verify that the result is a parallelogram.

Step 1: Find the midpoint of side AB.

M_1 = \left(\frac{0+6}{2},\;\frac{0+0}{2}\right) = (3,\,0)

Step 2: Find the midpoint of side BC.

M_2 = \left(\frac{6+8}{2},\;\frac{0+4}{2}\right) = (7,\,2)

Step 3: Find the midpoint of side CD.

M_3 = \left(\frac{8+2}{2},\;\frac{4+6}{2}\right) = (5,\,5)

Step 4: Find the midpoint of side DA.

M_4 = \left(\frac{2+0}{2},\;\frac{6+0}{2}\right) = (1,\,3)

Step 5: Verify it is a parallelogram by checking that opposite sides are parallel. Compute the vectors for opposite sides.

\vec{M_1M_2} = (7-3,\;2-0) = (4,\,2)$$ $$\vec{M_4M_3} = (5-1,\;5-3) = (4,\,2)$$ $$\vec{M_1M_4} = (1-3,\;3-0) = (-2,\,3)$$ $$\vec{M_2M_3} = (5-7,\;5-2) = (-2,\,3)

Answer: The Varignon parallelogram has vertices (3, 0), (7, 2), (5, 5), and (1, 3). Since opposite side vectors are equal — (4, 2) and (−2, 3) — the figure is indeed a parallelogram.

Another Example

This example focuses on the area relationship rather than just finding the vertices. It demonstrates the key theorem that the Varignon parallelogram always has half the area of the original quadrilateral.

Problem: Using the same quadrilateral A(0, 0), B(6, 0), C(8, 4), D(2, 6), verify that the area of the Varignon parallelogram is half the area of the original quadrilateral.

Step 1: Compute the area of quadrilateral ABCD using the shoelace formula.

\text{Area}_{ABCD} = \frac{1}{2}|x_A(y_B - y_D) + x_B(y_C - y_A) + x_C(y_D - y_B) + x_D(y_A - y_C)|$$ $$= \frac{1}{2}|0(0-6) + 6(4-0) + 8(6-0) + 2(0-4)|$$ $$= \frac{1}{2}|0 + 24 + 48 - 8| = \frac{1}{2}(64) = 32

Step 2: Compute the area of the Varignon parallelogram with vertices M₁(3,0), M₂(7,2), M₃(5,5), M₄(1,3) using the shoelace formula.

\text{Area}_{V} = \frac{1}{2}|3(2-3) + 7(5-0) + 5(3-2) + 1(0-5)|$$ $$= \frac{1}{2}|{-3} + 35 + 5 - 5| = \frac{1}{2}(32) = 16

Step 3: Compare the two areas.

\frac{\text{Area}_{V}}{\text{Area}_{ABCD}} = \frac{16}{32} = \frac{1}{2} \;\checkmark

Answer: The Varignon parallelogram has area 16, which is exactly half of the original quadrilateral's area of 32, confirming the half-area theorem.

Frequently Asked Questions

Does the Varignon parallelogram work for any quadrilateral?

Yes. Varignon's theorem guarantees that connecting the midpoints of consecutive sides of any quadrilateral — convex, concave, or even self-intersecting — always produces a parallelogram. The proof relies on the midpoint theorem applied to the two diagonals of the quadrilateral: each side of the inner figure is parallel to one of the diagonals and half its length.

Why is the Varignon parallelogram always exactly half the area?

Each side of the Varignon parallelogram is parallel to a diagonal of the original quadrilateral and half as long. When you connect the midpoints, the four corner triangles that are "cut off" collectively account for exactly half the area of the original quadrilateral, leaving the inner parallelogram with the other half. This result holds for all quadrilaterals, including non-convex ones (using signed area for self-intersecting cases).

When is the Varignon parallelogram a rectangle, rhombus, or square?

The Varignon parallelogram is a rectangle when the diagonals of the original quadrilateral are perpendicular (as in a kite). It is a rhombus when the diagonals of the original quadrilateral are equal in length (as in an isosceles trapezoid or a rectangle). It is a square when the original quadrilateral has diagonals that are both perpendicular and equal in length (as in a square).

Varignon Parallelogram vs. Original Quadrilateral

	Varignon Parallelogram	Original Quadrilateral
Shape guarantee	Always a parallelogram	Can be any shape (convex, concave, self-intersecting)
Area	Exactly ½ the area of the original	Full area
Side lengths	Each side = ½ the length of a diagonal of the original	Four arbitrary side lengths
Sides parallel to	The diagonals of the original quadrilateral	No required parallelism

Why It Matters

Varignon's theorem appears in geometry courses as a striking application of the midpoint theorem and vector methods. It shows that hidden structure (a parallelogram) exists inside every quadrilateral, no matter how irregular. Problems involving Varignon parallelograms also arise in competition mathematics and in proofs about quadrilateral area and diagonal relationships.

Common Mistakes

Mistake: Connecting midpoints of opposite sides instead of adjacent (consecutive) sides.

Correction: You must connect the midpoints of consecutive sides in order: midpoint of AB to midpoint of BC to midpoint of CD to midpoint of DA. Connecting midpoints of opposite sides gives you the two "bimedians" of the quadrilateral, which is a different construction.

Mistake: Assuming the Varignon parallelogram only works for convex quadrilaterals.

Correction: The theorem holds for all quadrilaterals — convex, concave, and even self-intersecting (crossed) ones. The proof depends only on the midpoint theorem applied to triangles formed by the diagonals, which has no convexity requirement.

Related Terms

Parallelogram — The shape always produced by the construction
Midpoint — Used to find the vertices of the Varignon parallelogram
Quadrilateral — The original figure whose midpoints are connected
Adjacent — Midpoints of adjacent (consecutive) sides are connected
Side of a Polygon — The segments whose midpoints are used
Diagonal — Each Varignon side is parallel to a diagonal and half its length
Shoelace Formula — Useful for computing areas to verify the half-area property
Midpoint Theorem — The key theorem underlying the proof of Varignon's result