Spieker center

In geometry, the Spieker center is a special point associated with a plane triangle. It is defined as the center of mass of the perimeter of the triangle. The Spieker center of a triangle Template:Math is the center of gravity of a homogeneous wire frame in the shape of Template:Math. The point is named in honor of the 19th-century German geometer Theodor Spieker. The Spieker center is a triangle center and it is listed as the point X(10) in Clark Kimberling's Encyclopedia of Triangle Centers.

Location

Construction of the Spieker center. Template:Legend Template:Legend Template:Legend-line Template:Legend-line

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

The Spieker center of triangle Template:Math is the incenter of the medial triangle of Template:Math.

That is, the Spieker center of Template:Math is the center of the circle inscribed in the medial triangle of Template:Math. This circle is known as the Spieker circle.

The Spieker center is also located at the intersection of the three cleavers of triangle Template:Math. 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 Template:Math, 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 Template:Math consisting of three wires in the form of line segments having lengths Template:Mvar. The wire frame has the same center of mass as a system of three particles of masses Template:Mvar placed at the midpoints Template:Mvar of the sides Template:Mvar. The centre of mass of the particles at Template:Mvar and Template:Mvar is the point Template:Mvar which divides the segment Template:Mvar in the ratio Template:Math. The line Template:Mvar is the internal bisector of Template:Math. The centre of mass of the three particle system thus lies on the internal bisector of Template:Math. Similar arguments show that the center mass of the three particle system lies on the internal bisectors of Template:Math and Template:Math 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 Template:Math , which is the incenter of the medial triangle Template:Math .

Properties

The Spieker center of a triangle is the cleavance center of the triangle. Template:Legend Template:Legend-line Template:Legend-line Template:Legend Template:Legend-line

Let Template:Mvar be the Spieker center of triangle Template:Math.

• The trilinear coordinates of Template:Mvar are

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

• The barycentric coordinates of Template:Mvar are

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

• Template:Mvar is the radical center of the three excircles.
• Template:Mvar is the cleavance center of triangle Template:Math
• Template:Mvar is collinear with the incenter (Template:Mvar), the centroid (Template:Mvar), and the Nagel point (Template:Mvar) of triangle Template:Math. Moreover,

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

Thus on a suitably scaled and positioned number line, Template:Math, Template:Math, Template:Math, and Template:Math.

• Template:Mvar lies on the Kiepert hyperbola. Template:Mvar is the point of concurrence of the lines Template:Mvar where Template:Math are similar, isosceles and similarly situated triangles constructed on the sides of triangle Template:Math as bases, having the common base angle

$\theta ={\tan }^{-1}[\tan \left(\frac{A}{2}\right)\tan \left(\frac{B}{2}\right)\tan \left(\frac{C}{2}\right)].$