**Bottema's theorem** is a theorem in plane geometry by the Dutch mathematician Oene Bottema (Groningen, 1901–1992).^{ [1] }

The theorem can be stated as follows: in any given triangle , construct squares on any two adjacent sides, for example and . The midpoint of the line segment that connects the vertices of the squares opposite the common vertex, *, of the two sides of the triangle is independent of the location of .*^{ [2] }

The theorem is true when the squares are constructed in one of the following ways:

- Looking at the figure, starting from the lower left vertex, , follow the triangle vertices clockwise and construct the squares to the left of the sides of the triangle.
- Follow the triangle in the same way and construct the squares to the right of the sides of the triangle.

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 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 geometry, an **altitude** of a triangle is a line segment through a vertex and perpendicular to a line containing the base. This line containing the opposite side is called the *extended base* of the altitude. The intersection of the extended base and the altitude is called the *foot* of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as *dropping the altitude* at that vertex. It is a special case of orthogonal projection.

In geometry, the **incircle** or **inscribed circle** of a triangle is the largest circle contained in the triangle; it touches the three sides. The center of the incircle is a triangle center called the triangle's incenter.

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 graph theory, **Turán's theorem** is a result on the number of edges in a *K*_{r+1}-free graph.

In mathematics, **Sperner's lemma** is a combinatorial analog of the Brouwer fixed point theorem, which is equivalent to it.

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 the mathematical discipline of graph theory, the **line graph** of an undirected graph *G* is another graph L(*G*) that represents the adjacencies between edges of *G*. L(*G*) is constructed in the following way: for each edge in *G*, make a vertex in L(*G*); for every two edges in *G* that have a vertex in common, make an edge between their corresponding vertices in L(*G*).

In plane geometry, **Morley's trisector theorem** states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the **first Morley triangle** or simply the **Morley triangle**. The theorem was discovered in 1899 by Anglo-American mathematician Frank Morley. It has various generalizations; in particular, if all of the trisectors are intersected, one obtains four other equilateral triangles.

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 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 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.

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 Euclidean geometry, the **isodynamic points** of a triangle are points associated with the triangle, with the properties that an inversion centered at one of these points transforms the given triangle into an equilateral triangle, and that the distances from the isodynamic point to the triangle vertices are inversely proportional to the opposite side lengths of the triangle. Triangles that are similar to each other have isodynamic points in corresponding locations in the plane, so the isodynamic points are triangle centers, and unlike other triangle centers the isodynamic points are also invariant under Möbius transformations. A triangle that is itself equilateral has a unique isodynamic point, at its centroid; every non-equilateral triangle has two isodynamic points. Isodynamic points were first studied and named by Joseph Neuberg (1885).

The **exterior angle theorem** is Proposition 1.16 in Euclid's Elements, which states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. This is a fundamental result in absolute geometry because its proof does not depend upon the parallel postulate.

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 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":

- ↑ Koetsier, T. (2007). "Oene Bottema (1901–1992)". In Ceccarelli, M. (ed.).
*Distinguished Figures in Mechanism and Machine Science*. History of Mechanism and Machine Science.**1**. Dordrecht: Springer. pp. 61–68. doi:10.1007/978-1-4020-6366-4_3. ISBN 978-1-4020-6365-7. - ↑ Shriki, A. (2011), "Back to Treasure Island",
*The Mathematics Teacher*,**104**(9): 658–664, JSTOR 20876991 .

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