Spieker center

In geometry, the Spieker center is a special point associated with a plane triangle.

[1][2] The point is named in honor of the 19th-century German geometer Theodor Spieker.

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 △ABC consisting of three wires in the form of line segments having lengths a, b, c. The wire frame has the same center of mass as a system of three particles of masses a, b, c placed at the midpoints D, E, F of the sides BC, CA, AB.

The centre of mass of the particles at E and F is the point P which divides the segment EF in the ratio c : b.

Similar arguments show that the center mass of the three particle system lies on the internal bisectors of ∠E and ∠F also.

Construction of the Spieker center.
Triangle △ ABC
Medial triangle △ DEF of △ ABC
Angle bisectors of △ DEF ( concurrent at the Spieker center S )
Inscribed circle of △ DEF ( Spieker circle of △ ABC ; centered at S )
The Spieker center of a triangle is the cleavance center of the triangle.
Triangle △ ABC
Angle bisectors of △ ABC (concurrent at the incenter I )
Cleavers of △ ABC (concurrent at the Spieker center S )
Medial triangle △ DEF of △ ABC
Inscribed circle of △ DEF (Spieker circle of △ ABC ; centered at S )