In geometry, the **Petr–Douglas–Neumann theorem** (or the **PDN-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.^{ [1] }^{ [2] } The theorem was independently rediscovered by Jesse Douglas (1897–1965) in 1940^{ [3] } and also by B H Neumann (1909–2002) in 1941.^{ [2] }^{ [4] } The naming of the theorem as *Petr–Douglas–Neumann theorem*, or as the *PDN-theorem* for short, is due to Stephen B Gray.^{ [2] } This theorem has also been called **Douglas's theorem**, the **Douglas–Neumann theorem**, the **Napoleon–Douglas–Neumann theorem** and **Petr's theorem**.^{ [2] }

- Statement of the theorem
- Specialisation to triangles
- Specialisation to quadrilaterals
- Construct A1 using apex angle π/2 and then A2 with apex angle π.
- Construct A1 using apex angle π and then A2 with apex angle π/2.
- Images illustrating application of the theorem to quadrilaterals
- Specialisation to pentagons
- Proof of the theorem
- References

The PDN-theorem is a generalisation of the Napoleon's theorem which is concerned about arbitrary triangles and of the van Aubel's theorem which is related to arbitrary quadrilaterals.

The Petr–Douglas–Neumann theorem asserts the following.^{ [3] }^{ [5] }

*If isosceles triangles with apex angles 2kπ/n are erected on the sides of an arbitrary n-gon A*._{0}, and if this process is repeated with the n-gon formed by the free apices of the triangles, but with a different value of k, and so on until all values 1 ≤ k ≤ n − 2 have been used (in arbitrary order), then a regular n-gon A_{n−2}is formed whose centroid coincides with the centroid of A_{0}

In the case of triangles, the value of *n* is 3 and that of *n* − 2 is 1. Hence there is only one possible value for *k*, namely 1. The specialisation of the theorem to triangles asserts that the triangle A_{1} is a regular 3-gon, that is, an equilateral triangle.

A_{1} is formed by the apices of the isosceles triangles with apex angle 2π/3 erected over the sides of the triangle A_{0}. The vertices of A_{1} are the centers of equilateral triangles erected over the sides of triangle A_{0}. Thus the specialisation of the PDN theorem to a triangle can be formulated as follows:

*If equilateral triangles are erected over the sides of any triangle, then the triangle formed by the centers of the three equilateral triangles is equilateral.*

The last statement is the assertion of the Napoleon's theorem.

In the case of quadrilaterals, the value of *n* is 4 and that of *n* − 2 is 2. There are two possible values for *k*, namely 1 and 2, and so two possible apex angles, namely:

- (2×1×π)/4 = π/2 = 90° ( corresponding to
*k*= 1 ) - (2×2×π)/4 = π = 180° ( corresponding to
*k*= 2 ).

According to the PDN-theorem the quadrilateral A_{2} is a regular 4-gon, that is, a square. The two-stage process yielding the square A_{2} can be carried out in two different ways. (The apex *Z* of an isosceles triangle with apex angle π erected over a line segment *XY* is the midpoint of the line segment *XY*.)

In this case the vertices of A_{1} are the free apices of isosceles triangles with apex angles π/2 erected over the sides of the quadrilateral A_{0}. The vertices of the quadrilateral A_{2} are the midpoints of the sides of the quadrilateral A_{1}. By the PDN theorem, A_{2} is a square.

The vertices of the quadrilateral A_{1} are the centers of squares erected over the sides of the quadrilateral A_{0}. The assertion that quadrilateral A_{2} is a square is equivalent to the assertion that the diagonals of A_{1} are equal and perpendicular to each other. The latter assertion is the content of van Aubel's theorem.

Thus van Aubel's theorem is a special case of the PDN-theorem.

In this case the vertices of A_{1} are the midpoints of the sides of the quadrilateral A_{0} and those of A_{2} are the apices of the triangles with apex angles π/2 erected over the sides of A_{1}. The PDN-theorem asserts that A_{2} is a square in this case also.

Diagram illustrating the fact that van Aubel's theorem is a special case of Petr–Douglas–Neumann theorem. |

In the case of pentagons, we have *n* = 5 and *n* − 2 = 3. So there are three possible values for *k*, namely 1, 2 and 3, and hence three possible apex angles for isosceles triangles:

- (2×1×π)/5 = 2π/5 = 72°
- (2×2×π)/5 = 4π/5 = 144°
- (2×3×π)/5 = 6π/5 = 216°

According to the PDN-theorem, A_{3} is a regular pentagon. The three-stage process leading to the construction of the regular pentagon A_{3} can be performed in six different ways depending on the order in which the apex angles are selected for the construction of the isosceles triangles.

Serial number | Apex angle in the construction of A _{1} | Apex angle in the construction of A _{2} | Apex angle in the construction of A _{3} |
---|---|---|---|

1 | 72° | 144° | 216° |

2 | 72° | 216° | 144° |

3 | 144° | 72° | 216° |

4 | 144° | 216° | 72° |

5 | 216° | 72° | 144° |

6 | 216° | 144° | 72° |

The theorem can be proved using some elementary concepts from linear algebra.^{ [2] }^{ [6] }

The proof begins by encoding an *n*-gon by a list complex numbers representing the vertices of the *n*-gon. This list can be thought of as a vector in the *n*-dimensional complex linear space C^{n}. Take an *n*-gon *A* and let it be represented by the complex vector

*A*= (*a*_{1},*a*_{2}, ... ,*a*_{n}).

Let the polygon *B* be formed by the free vertices of similar triangles built on the sides of *A* and let it be represented by the complex vector

*B*= (*b*_{1},*b*_{2}, ... ,*b*_{n}).

Then we have

- α(
*a*_{r}−*b*_{r}) =*a*_{r+1}−*b*_{r}, where α = exp(*i*θ ) for some θ (here*i*is the square root of −1).

This yields the following expression to compute the *b*_{r} ' s:

*b*_{r}= (1−α)^{−1}(*a*_{r+1}− α*a*_{r}).

In terms of the linear operator *S* : C^{n} → C^{n} that cyclically permutes the coordinates one place, we have

*B*= (1−α)^{−1}(*S*− α*I*)*A*, where*I*is the identity matrix.

This means that the polygon *A*_{n−2} that we need to show is regular is obtained from *A*_{0} by applying the composition of the following operators:

- ( 1 − ω
^{k})^{−1}(*S*− ω^{k}*I*) for*k*= 1, 2, ... ,*n*− 2, where ω = exp( 2π*i*/*n*). (These commute because they are all polynomials in the same operator*S*.)

A polygon *P* = ( *p*_{1}, *p*_{2}, ..., *p*_{n} ) is a regular *n*-gon if each side of *P* is obtained from the next by rotating through an angle of 2π/*n*, that is, if

*p*_{r + 1}−*p*_{r}= ω(*p*_{r + 2}−*p*_{r + 1}).

This condition can be formulated in terms of S as follows:

- (
*S*−*I*)(*I*− ω*S*)*P*= 0.

Or equivalently as

- (
*S*−*I*)(*S*− ω^{n− 1}*I*)*P*= 0, since ω^{n}*= 1.*

Petr–Douglas–Neumann theorem now follows from the following computations.

- (
*S*−*I*)(*S*− ω^{n− 1}*I*)*A*_{n− 2}- = (
*S*−*I*)(*S*− ω^{n− 1}*I*) ( 1 − ω )^{−1}(*S*− ω*I*) ( 1 − ω^{2})^{−1}(*S*− ω^{2}*I*) ... ( 1 − ω^{n− 2})^{−1}(*S*− ω^{n− 2}*I*)*A*_{0} - = ( 1 − ω )
^{−1}( 1 − ω^{2})^{−1}... ( 1 − ω^{n− 2})^{−1}(*S*−*I*) (*S*−ω*I*) (*S*−ω^{2}*I*) ... (*S*−ω^{n− 1}*I*)*A*_{0} - = ( 1 − ω )
^{−1}( 1 − ω^{2})^{−1}... ( 1 − ω^{n− 2})^{−1}(*S*^{n}−*I*)*A*_{0} - = 0, since
*S*^{n}=*I*.

- = (

In geometry, an ** n-gonal antiprism** or

An *n*-gonal **bipyramid** or **dipyramid** is a polyhedron formed by joining an *n*-gonal pyramid and its mirror image base-to-base. An *n*-gonal bipyramid has 2*n* triangle faces, 3*n* edges, and 2 + *n* vertices.

In geometry, a **polygon** is a plane figure that is described by a finite number of straight line segments connected to form a closed polygonal chain or *polygonal circuit*. The solid plane region, the bounding circuit, or the two together, may be called a **polygon**.

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 .

A **triangle** is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices *A*, *B*, and *C* is denoted .

In geometry, a **hexagon** is a six-sided polygon or 6-gon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.

In Euclidean geometry, a **kite ** is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite to each other instead of being adjacent. Kite quadrilaterals are named for the wind-blown, flying kites, which often have this shape and which are in turn named for a bird. Kites are also known as **deltoids**, but the word "deltoid" may also refer to a deltoid curve, an unrelated geometric object.

In geometry, an **equilateral triangle** is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a **regular triangle**.

In geometry, three or more than three straight lines make a polygon and an **equilateral polygon** is a polygon which has all sides of the same length. Except in the triangle case, it doesn’t need to be equiangular, but if it does then it is a regular polygon. If the number of sides is at least five, an equilateral polygon doesn’t need to be a convex polygon: it could be concave or even self-intersecting.

In geometry, an **isosceles triangle** is a triangle that has two sides of equal length. Sometimes it is specified as having *exactly* two sides of equal length, and sometimes as having *at least* two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids.

In Euclidean geometry, a **regular polygon** is a polygon that is equiangular and equilateral. Regular polygons may be either **convex** or **star**. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.

In geometry, the **midpoint** is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.

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*.

In geometry, a **cupola** is a solid formed by joining two polygons, one with twice as many edges as the other, by an alternating band of isosceles triangles and rectangles. If the triangles are equilateral and the rectangles are squares, while the base and its opposite face are regular polygons, the triangular, square, and pentagonal cupolae all count among the Johnson solids, and can be formed by taking sections of the cuboctahedron, rhombicuboctahedron, and rhombicosidodecahedron, respectively.

In geometry, the **circumscribed circle** or **circumcircle** of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the **circumcenter** and its radius is called the **circumradius**.

In Euclidean geometry, an **equiangular polygon** is a polygon whose vertex angles are equal. If the lengths of the sides are also equal then it is a regular polygon. Isogonal polygons are equiangular polygons which alternate two edge lengths.

In geometry, the **Fermat point** of a triangle, also called the **Torricelli point** or **Fermat–Torricelli point**, is a point such that the total distance from the three vertices of the triangle to the point is the minimum possible. It is so named because this problem was first raised by Fermat in a private letter to Evangelista Torricelli, who solved it.

In geometry, a **pyramid** is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a *lateral face*. It is a conic solid with polygonal base. A pyramid with an *n*-sided base has *n* + 1 vertices, *n* + 1 faces, and 2*n* edges. All pyramids are self-dual.

In geometry, the **biggest little polygon** for a number *n* is the *n*-sided polygon that has diameter one and that has the largest area among all diameter-one *n*-gons. One non-unique solution when *n* = 4 is a square, and the solution is a regular polygon when *n* is an odd number, but the solution is irregular otherwise.

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.

- ↑ K. Petr (1908). "Ein Satz ¨uber Vielecke".
*Arch. Math. Phys*.**13**: 29–31. - 1 2 3 4 5 Stephen B. Gray (2003). "Generalizing the Petr–Douglas–Neumann Theorem on
*n*-gons" (PDF).*American Mathematical Monthly*.**110**(3): 210–227. CiteSeerX 10.1.1.605.2676 . doi:10.2307/3647935. JSTOR 3647935 . Retrieved 8 May 2012. - 1 2 Douglas, Jesse (1946). "On linear polygon transformations" (PDF).
*Bulletin of the American Mathematical Society*.**46**(6): 551–561. doi:10.1090/s0002-9904-1940-07259-3 . Retrieved 7 May 2012. - ↑ B H Neumann (1941). "Some remarks on polygons".
*Journal of the London Mathematical Society*. s1-16 (4): 230–245. doi:10.1112/jlms/s1-16.4.230 . Retrieved 7 May 2012. - ↑ van Lamoen, Floor; Weisstein, Eric W. "Petr–Neumann–Douglas Theorem". From MathWorld—A Wolfram Web Resource. Retrieved 8 May 2012.
- ↑ Omar Antolín Camarena. "The Petr–Neumann–Douglas theorem through linear algebra" . Retrieved 10 Jan 2018.

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