In geometry, the Spieker center is a special point associated with a plane triangle. 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 of the triangle. The Spieker center of a triangle is the center of gravity of a homogeneous wire frame in the shape of . The point is named in honor of the 19th-century German geometer

Theodor Spieker Theodor Spieker (8 August 1823 – 9 April 1913) was a German mathematician, a teacher at a gymnasium in Potsdam. Spieker's geometry textbook (Verlag von August Stein, Potsdam, 1862) was republished in many editions. A copy of this textbook was ...

. The Spieker center is a

triangle center In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For examp ...

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 s ...

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 bise ...

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 no ...

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

of . This circle is known as the

Spieker circle In geometry, the incircle of the medial triangle of a triangle is the Spieker circle, named after 19th-century German geometer Theodor Spieker. Its center, the Spieker center, in addition to being the incenter of the medial triangle, is the ...

. 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 ...

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 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 concurre ...

() 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