In plane geometry, **Van Aubel's theorem** describes a relationship between squares constructed on the sides of a quadrilateral. Starting with a given convex quadrilateral, construct a square, external to the quadrilateral, on each side. Van Aubel's theorem states that the two line segments between the centers of opposite squares are of equal lengths and are at right angles to one another. Another way of saying the same thing is that the center points of the four squares form the vertices of an equidiagonal orthodiagonal quadrilateral. The theorem is named after H. H. van Aubel, who published it in 1878.^{ [1] }

The theorem holds true also for re-entrant quadrilaterals,^{ [2] } and when the squares are constructed internal to the given quadrilateral.^{ [3] } For complex (self-intersecting) quadrilaterals, the *external* and *internal* constructions for the squares are not definable. In this case, the theorem holds true when the constructions are carried out in the more general way:^{ [3] }

- follow the quadrilateral vertexes in a sequential direction and construct each square on the right hand side of each side of the given quadrilateral.
- Follow the quadrilateral vertexes in the same sequential direction and construct each square on the left hand side of each side of the given quadrilateral.

A few extensions of the theorem, considering similar rectangles, similar rhombi and similar parallelograms constructed on the sides of the given quadrilateral, have been published on the The Mathematical Gazette.^{ [4] }^{ [5] }

In Euclidean plane geometry, a **quadrilateral** is a polygon with four edges (sides) and four vertices (corners). Other names for quadrilateral include **quadrangle**, **tetragon**, and **4-gon**. A quadrilateral with vertices , , and is sometimes denoted as .

In Euclidean plane geometry, a **rectangle** is a quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal. It can also be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term **oblong** is occasionally used to refer to a non-square rectangle. A rectangle with vertices *ABCD* would be denoted as *ABCD*.

In elementary geometry, the property of being **perpendicular** (**perpendicularity**) is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects.

In Euclidean geometry, a **parallelogram** is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

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In Euclidean geometry, a **cyclic quadrilateral** or **inscribed quadrilateral** is a quadrilateral whose vertices all lie on a single circle. This circle is called the *circumcircle* or circumscribed circle, and the vertices are said to be *concyclic*. The center of the circle and its radius are called the *circumcenter* and the *circumradius* respectively. Other names for these quadrilaterals are **concyclic quadrilateral** and **chordal quadrilateral**, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

In geometry, a **square** is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices *ABCD* would be denoted *ABCD*.

Lines in a plane or higher-dimensional space are said to be **concurrent** if they intersect at a single point.

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In geometry, given a triangle *ABC* and a point *P* on its circumcircle, the three closest points to *P* on lines *AB*, *AC*, and *BC* are collinear. The line through these points is the **Simson line** of *P*, named for Robert Simson. The concept was first published, however, by William Wallace in 1799.

**Varignon's theorem** is a statement in Euclidean geometry, that deals with the construction of a particular parallelogram, the **Varignon parallelogram**, from an arbitrary quadrilateral (quadrangle). It is named after Pierre Varignon, whose proof was published posthumously in 1731.

**Miquel's theorem** is a result in geometry, named after Auguste Miquel, concerning the intersection of three circles, each drawn through one vertex of a triangle and two points on its adjacent sides. It is one of several results concerning circles in Euclidean geometry due to Miquel, whose work was published in Liouville's newly founded journal *Journal de mathématiques pures et appliquées*.

In mathematics, the **Pythagorean theorem**, also known as **Pythagoras's theorem**, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides *a*, *b* and *c*, often called the "Pythagorean equation":

The **Finsler–Hadwiger theorem** is statement in Euclidean plane geometry that describes a third square derived from any two squares that share a vertex. The theorem is named after Paul Finsler and Hugo Hadwiger, who published it in 1937 as part of the same paper in which they published the Hadwiger–Finsler inequality relating the side lengths and area of a triangle.

In Euclidean geometry, an **ex-tangential quadrilateral** is a convex quadrilateral where the *extensions* of all four sides are tangent to a circle outside the quadrilateral. It has also been called an **exscriptible quadrilateral**. The circle is called its *excircle*, its radius the *exradius* and its center the *excenter*. The excenter lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect. The ex-tangential quadrilateral is closely related to the tangential quadrilateral.

In Euclidean geometry, an **equidiagonal quadrilateral** is a convex quadrilateral whose two diagonals have equal length. Equidiagonal quadrilaterals were important in ancient Indian mathematics, where quadrilaterals were classified first according to whether they were equidiagonal and then into more specialized types.

In geometry, **Napoleon points** are a pair of special points associated with a plane triangle. It is generally believed that the existence of these points was discovered by Napoleon Bonaparte, the Emperor of the French from 1804 to 1815, but many have questioned this belief. The Napoleon points are triangle centers and they are listed as the points X(17) and X(18) in Clark Kimberling's Encyclopedia of Triangle Centers.

In geometry, the **Petr–Douglas–Neumann theorem** is a result concerning arbitrary planar polygons. The theorem asserts that a certain procedure when applied to an arbitrary polygon always yields a regular polygon having the same number of sides as the initial polygon. The theorem was first published by Karel Petr (1868–1950) of Prague in 1908. The theorem was independently rediscovered by Jesse Douglas (1897–1965) in 1940 and also by B H Neumann (1909–2002) in 1941. The naming of the theorem as *Petr–Douglas–Neumann theorem*, or as the *PDN-theorem* for short, is due to Stephen B Gray. This theorem has also been called **Douglas's theorem**, the **Douglas–Neumann theorem**, the **Napoleon–Douglas–Neumann theorem** and **Petr's theorem**.

**Bottema's theorem** is a theorem in plane geometry by the Dutch mathematician Oene Bottema.

- ↑ van Aubel, H. H. (1878), "Note concernant les centres de carrés construits sur les côtés d'un polygon quelconque",
*Nouvelle Correspondance Mathématique*(in French),**4**: 40–44. - ↑ Coxeter, H.S.M., and Greitzer, Samuel L. 1967.
*Geometry Revisited*, pages 52. - 1 2 D. Pellegrinetti: "The Six-Point Circle for the Quadrangle".
*International Journal of Geometry*, Vol. 8 (Oct., 2019), No. 2, pp. 5–13. - ↑ M. de Villiers: "Dual Generalizations of Van Aubel's theorem".
*The Mathematical Gazette*, Vol. 82 (Nov., 1998), pp. 405-412. - ↑ J. R. Silvester: "Extensions of a Theorem of Van Aubel".
*The Mathematical Gazette*, Vol. 90 (Mar., 2006), pp. 2-12.

Wikimedia Commons has media related to . Van Aubel's theorem |

- Weisstein, Eric W. "van Aubel's Theorem".
*MathWorld*. - Van Aubel's Theorem for Quadrilaterals and Van Aubel's Theorem for Triangles by Jay Warendorff, The Wolfram Demonstrations Project.
- The Beautiful Geometric Theorem of Van Aubel by Yutaka Nishiyama, International Journal of Pure and Applied Mathematics.
- Interactive applet by Tim Brzezinski showing Van Aubel's Theorem made using GeoGebra.
- Some generalizations of Van Aubel's theorem to similar quadrilaterals at Dynamic Geometry Sketches, interactive geometry sketches.

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