Bottema's theorem

Last updated
Bottema's theorem construction; changing the location of vertex
C
{\textstyle C}
changes the locations of vertices
E
{\textstyle E}
and
F
{\textstyle F}
but does not change the location of their midpoint
M
{\textstyle M} Bottema's theorem.png
Bottema's theorem construction; changing the location of vertex changes the locations of vertices and but does not change the location of their midpoint

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

Contents

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:

See also

Related Research Articles

Quadrilateral polygon with four sides and four corners

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 .

Triangle Shape with three sides

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 .

Hexagon Shape with a few sides

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°.

Rectangle Quadrilateral with four right angles

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.

Altitude (triangle)

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.

Incircle and excircles of a triangle Circles tangent to all three sides of a triangle

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.

Equilateral triangle

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 Kr+1-free graph.

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

Square Regular quadrilateral

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).

Morleys trisector theorem 3 intersections of any triangles adjacent angle trisectors form an equilateral triangle

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.

Circumscribed circle

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.

Fermat point

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.

Van Aubels theorem Lines connecting the centers of squares on opposite sides of a quadrilateral are = and ⟂

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.

Pyramid (geometry) Conic solid with a polygonal base

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 2n edges. All pyramids are self-dual.

Isodynamic point

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.

Pentagon shape with five sides

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

Pythagorean theorem Equation relating the side lengths of a right triangle

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":

References

  1. 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.
  2. Shriki, A. (2011), "Back to Treasure Island", The Mathematics Teacher , 104 (9): 658–664, JSTOR   20876991 .