VARIGNON'S THEOREM: EXAMPLES AND SOLVED EXERCISES - MATHS - 2025

The theorem Varignon states that if any quadrilateral are continuously connected midpoints of the sides, a parallelogram is generated. This theorem was formulated by Pierre Varignon and published in 1731 in the book Elements of mathematics ”.

The publication of the book occurred years after his death. Since it was Varignon who introduced this theorem, the parallelogram is named after him. The theorem is based on Euclidean geometry and presents geometric relationships of the quadrilaterals.

What is Varignon's theorem?

Varignon stated that a figure that is defined by the midpoints of a quadrilateral will always result in a parallelogram, and the area of ​​the parallelogram will always be half the area of ​​the quadrilateral if it is flat and convex. For example:

In the figure you can see a quadrilateral with an area X, where the midpoints of the sides are represented by E, F, G and H and, when joined, form a parallelogram. The area of ​​the quadrilateral will be the sum of the areas of the triangles that are formed, and half of this corresponds to the area of ​​the parallelogram.

Since the area of ​​the parallelogram is half the area of ​​the quadrilateral, the perimeter of that parallelogram can be determined.

Thus, the perimeter is equal to the sum of the lengths of the diagonals of the quadrilateral; this is because the medians of the quadrilateral will be the diagonals of the parallelogram.

On the other hand, if the lengths of the diagonals of the quadrilateral are exactly the same, the parallelogram will be a rhombus. For example:

From the figure it can be seen that, by joining the midpoints of the sides of the quadrilateral, a rhombus is obtained. On the other hand, if the diagonals of the quadrilateral are perpendicular, the parallelogram will be a rectangle.

Also the parallelogram will be a square when the quadrilateral has the diagonals with the same length and they are also perpendicular.

The theorem is not only fulfilled in plane quadrilaterals, it is also implemented in spatial geometry or in large dimensions; that is, in those quadrilaterals that are not convex. An example of this can be an octahedron, where the midpoints are the centroids of each face and form a parallelepiped.

In this way, by joining the midpoints of different figures, parallelograms can be obtained. An easy way to check if this is really true is that the opposite sides must be parallel when extended.

Examples

First example

Extension of opposite sides to show that it is a parallelogram:

Second example

By joining the midpoints of a rhombus, a rectangle is obtained:

The theorem is used in the union of points located in the middle of the sides of a quadrilateral, and it can also be used for other types of points, such as a trisection, penta-section, or even an infinite number of sections (nth), in order to divide the sides of any quadrilateral into segments that are proportional.

Solved exercises

Exercise 1

In the figure we have a quadrilateral ABCD of area Z, where the midpoints of the sides of this are PQSR. Check that a Varignon parallelogram is formed.

Solution

It can be seen that by joining the PQSR points a Varignon parallelogram is formed, precisely because the midpoints of a quadrilateral are given in the statement.

To demonstrate this, first the midpoints PQSR are joined, so it can be seen that another quadrilateral is formed. To show that it is a parallelogram, you only have to draw a straight line from point C to point A, so it can be seen that CA is parallel to PQ and RS.

In the same way, when extending the sides PQRS it can be seen that PQ and RS are parallel, as shown in the following image:

Exercise 2

We have a rectangle such that the lengths of all its sides are equal. By joining the midpoints of these sides, a rhombus ABCD is formed, which is divided by two diagonals AC = 7cm and BD = 10cm, which coincide with the measurements of the sides of the rectangle. Determine the areas of the rhombus and the rectangle.

Solution

Remembering that the area of ​​the resulting parallelogram is half of the quadrilateral, the area of ​​these can be determined knowing that the measure of the diagonals coincides with the sides of the rectangle. So you have to:

AB = D

CD = d

A _rectangle = (AB _* CD) = (10cm _* 7cm) = 70cm ²

A _rhombus = A _rectangle / 2

A _rhombus = 70 cm ² /2 = 35 cm ²

Exercise 3

In the figure there is a quadrilateral that has the union of the points EFGH, the lengths of the segments are given. Determine if the union of EFGH is a parallelogram.

AB = 2.4 CG = 3.06

EB = 1.75 GD = 2.24

BF = 2.88 DH = 2.02

HR = 3.94 HA = 2.77

Solution

As the lengths of the segments are given, it can be verified if there is proportionality between the segments; that is, you can know if they are parallel, relating the segments of the quadrilateral as follows:

- AE / EB = 2.4 / 1.75 = 1.37

- AH / HD = 2.77 / 2.02 = 1.37

- CF / FB = 3.94 / 2.88 = 1.37

- CG / GD = 3.06 / 2.24 = 1.37

Then the proportionality is checked, since:

AE / EB = AH / HD = CF / FB = CG / GD

Similarly, when drawing a line from point B to point D, it can be seen that EH is parallel to BD, just as BD is parallel to FG. On the other hand, EF is parallel to GH.

Thus it can be determined that EFGH is a parallelogram, because the opposite sides are parallel.

References

1. Andres, T. (2010). Mathematical Olympiad Tresure. Springer. New York.
2. Barbosa, JL (2006). Plane Euclidean Geometry. SBM. Rio de Janeiro.
3. Howar, E. (1969). Study of Geometries. Mexico: Hispanic - American.
4. Ramo, GP (1998). Unknown solutions to the Fermat-Torricelli problems. ISBN - Independent work.
5. Vera, F. (1943). Elements of Geometry. Bogota
6. Villiers, M. (1996). Some Adventures in Euclidean Geometry. South Africa.