geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the Spieker center is a special point associated with a plane

triangle 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 \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

. It is defined as the

center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...

of the

perimeter A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several pr ...

of the triangle. The Spieker center of a triangle is the

center of gravity In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force ma ...

of a homogeneous wire frame in the shape of . The point is named in honor of the 19th-century

German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ge ...

geometer A geometer is a mathematician whose area of study is geometry. Some notable geometers and their main fields of work, chronologically listed, are: 1000 BCE to 1 BCE * Baudhayana (fl. c. 800 BC) – Euclidean geometry, geometric algebra * ...

Theodor Spieker. The Spieker center is a triangle center and it is listed as the point ''X''(10) in

Clark Kimberling Clark Kimberling (born November 7, 1942 in Hinsdale, Illinois) is a mathematician, musician, and composer. He has been a mathematics professor since 1970 at the University of Evansville. His research interests include triangle centers, integer ...

Encyclopedia of Triangle Centers The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or "centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville. , the ...

Location

The following result can be used to locate the Spieker center of any triangle. :The Spieker center of triangle is the

incenter In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bis ...

of the

medial triangle In Euclidean geometry, the medial triangle or midpoint triangle of a triangle is the triangle with vertices at the midpoints of the triangle's sides . It is the case of the midpoint polygon of a polygon with sides. The medial triangle is n ...

of . That is, the Spieker center of is the center of the circle

inscribed {{unreferenced, date=August 2012 An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figu ...

in the

of . This circle is known as the Spieker circle. The Spieker center is also located at the intersection of the three

cleavers ''Galium aparine'', with common names including cleavers, clivers, catchweed and sticky willy among others, is an annual, herbaceous plant of the family Rubiaceae. Names ''Galium aparine'' is known by a variety of common names in English. They ...

of triangle . A cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. Each cleaver contains the center of mass of the boundary of , so the three cleavers meet at the Spieker center. To see that the incenter of the medial triangle coincides with the intersection point of the cleavers, consider a homogeneous wireframe in the shape of triangle consisting of three wires in the form of line segments having lengths . The wire frame has the same center of mass as a system of three particles of masses placed at the midpoints of the sides . The centre of mass of the particles at and is the point which divides the segment in the ratio . The line is the internal bisector of . The centre of mass of the three particle system thus lies on the internal bisector of . Similar arguments show that the center mass of the three particle system lies on the internal bisectors of and also. It follows that the center of mass of the wire frame is the point of concurrence of the internal bisectors of the angles of the triangle , which is the incenter of the medial triangle .

Properties

Let be the Spieker center of triangle . *The

trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...

of are ::

bc(b+c) : ca(c+a) : ab(a+b).

*The barycentric coordinates of are ::

b+c : c+a : a+b.

* is the

radical center The term radical center can refer to: * Radical centrism, a political movement * a mathematical construct: also called the power center (geometry) In geometry, the power center of three circles, also called the radical center, is the interse ...

of the three

excircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

s. * is the cleavance center of triangle * is

collinear 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 (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...

with the

(), the

centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...

(), and the

Nagel point In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concu ...

() of triangle . Moreover, ::

IS= SM, \quad IG= 2 \cdot GS, \quad MG= 2\cdot IG.

:Thus on a suitably scaled and positioned number line, , , , and . * lies on the Kiepert hyperbola. is the point of concurrence of the lines where are similar, isosceles and similarly situated triangles constructed on the sides of triangle as bases, having the common base angle ::

\theta = \tan^\left \tan\left(\frac\right) \tan\left(\frac\right) \tan\left(\frac\right) \right

References

{{reflist Triangle centers