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**Thomsen's theorem**, named after Gerhard Thomsen, is a theorem in elementary geometry. It shows that a certain path constructed by line segments being parallel to the edges of a triangle always ends up at its starting point.

**Gerhard Thomsen** was a German mathematician, probably best known for his work in various branches of geometry.

In geometry, a **polygonal chain** is a connected series of line segments. More formally, a polygonal chain *P* is a curve specified by a sequence of points called its vertices. The curve itself consists of the line segments connecting the consecutive vertices.

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 .

Consider an arbitrary triangle *ABC* with a point *P*_{1} on its edge *BC*. A sequence of points and parallel lines is constructed as follows. The parallel line to *AC* through *P*_{1} intersects *AB* in *P*_{2} and the parallel line to BC through *P*_{2} intersects AC in *P*_{3}. Continuing in this fashion the parallel line to AB through *P*_{3} intersects BC in *P*_{4} and the parallel line to *AC* through *P*_{4} intersects *AB* in *P*_{5}. Finally the parallel line to *BC* through *P*_{5} intersects AC in *P*_{6} and the parallel line to *AB* through *P*_{6} intersects *BC* in *P*_{7}. Thomsen's theorem now states that *P*_{7} is identical to *P*_{1} and hence the construction always leads to a closed path *P*_{1}*P*_{2}*P*_{3}*P*_{4}*P*_{5}*P*_{6}*P*_{1}

In modern mathematics, a **point** refers usually to an element of some set called a space.

In geometry, an intersection is a point, line, or curve common to two or more objects. The simplest case in Euclidean geometry is the intersection of two distinct lines, which either is one point or does not exist if the lines are parallel.

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 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 geometry, **bisection** is the division of something into two equal or congruent parts, usually by a line, which is then called a *bisector*. The most often considered types of bisectors are the *segment bisector* and the *angle bisector*.

In geometry, **Thales' theorem** states that if A, B, and C are distinct points on a circle where the line AC is a diameter, then the angle ∠ABC is a right angle. Thales' theorem is a special case of the inscribed angle theorem, and is mentioned and proved as part of the 31st proposition, in the third book of Euclid's *Elements*. It is generally attributed to Thales of Miletus, who is said to have offered an ox as a sacrifice of thanksgiving for the discovery, but sometimes it is attributed to Pythagoras.

In projective geometry, **Desargues's theorem**, named after Girard Desargues, states:

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 mathematics, the **Mohr–Mascheroni theorem** states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone.

In hyperbolic geometry, the **ultraparallel theorem** states that every pair of ultraparallel lines has a unique common perpendicular hyperbolic line.

In geometry, the **angle bisector theorem** is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle.

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 is first raised by Fermat in a private letter to Evangelista Torricelli, who solved it.

In geometry, **collinearity** of a set of points is the property of their lying on a single line. A set of points with this property is said to be **collinear**. In greater generality, the term has been used for aligned objects, that is, things being "in a line" or "in a row".

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.

In geometry, **Pasch's axiom** is a statement in plane geometry, used implicitly by Euclid, which cannot be derived from the postulates as Euclid gave them. Its essential role was discovered by Moritz Pasch in 1882.

The **intercept theorem**, also known as **Thales' theorem** or **basic proportionality theorem**, is an important theorem in elementary geometry about the ratios of various line segments that are created if two intersecting lines are intercepted by a pair of parallels. It is equivalent to the theorem about ratios in similar triangles. Traditionally it is attributed to Greek mathematician Thales.

**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 trigonometry, the **law of cosines** relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states

In geometry, a **pentagon** is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

In geometry the **Gossard perspector** is a special point associated with a plane triangle. It is a triangle center and it is designated as X(402) in Clark Kimberling's Encyclopedia of Triangle Centers. The point was named *Gossard perspector* by John Conway in 1998 in honour of Harry Clinton Gossard who discovered its existence in 1916. Later it was learned that the point had appeared in an article by Christopher Zeeman published during 1899 – 1902. From 2003 onwards the Encyclopedia of Triangle Centers has been referring to this point as *Zeeman–Gossard perspector*.

In Euclidean geometry, the **Droz-Farny line theorem** is a property of two perpendicular lines through the orthocenter of an arbitrary triangle.

Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate, is changed. The fifth axiom of hyperbolic geometry says that given a line *L* and a point *P* not on that line, there are at least two lines passing through *P* that are parallel to *L*. As in Euclidean geometry, where ancient Greek mathematicians used a compass and idealized ruler for constructions of lengths, angles, and other geometric figures, constructions can also be made in hyperbolic geometry.

*Satz von Thomsen*In:*Schülerduden – Mathematik II*. Bibliographisches Institut & F. A. Brockhaus, 2004, ISBN 3-411-04275-3, pp. 358–359 (German)

The **International Standard Book Number** (**ISBN**) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

- Darij Grinberg:
*Schließungssätze in der ebenen Geometrie*(German) - Weisstein, Eric W. "Thomsen's Figure".
*MathWorld*.

**Eric Wolfgang Weisstein** is an encyclopedist who created and maintains *MathWorld* and *Eric Weisstein's World of Science* (*ScienceWorld*). He is the author of the *CRC Concise Encyclopedia of Mathematics*. He currently works for Wolfram Research, Inc.

* MathWorld* is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana–Champaign.

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