Spieker circle

In geometry, the incircle of the medial triangle of a triangle ABC is the Spieker circle. Its center, the Spieker center, is the center of mass of the boundary of triangle ABC as well as being the incenter of the medial triangle.

The Spieker center (X₁₀ according to Kimberling's enumeration) is collinear with the incenter (X₁), centroid (X₂), and Nagel point (X₈). Indeed, the distances between pairs of the four points have constant ratios, regardless of the shape of triangle ABC, as indicated here:

$\displaystyle |X_{8}X_{10}|:|X_{8}X_{2}|:|X_{8}X_{1}|=3:4:6$

See also

Triangle center

References

• Johnson, Roger A. (1929). Modern Geometry. Boston: Houghton Mifflin. Dover reprint, 1960.

• Kimberling, Clark (1998). "Triangle centers and central triangles". Congressus Numerantium. 129: i–xxv, 1–295.

External links

• Spieker Conic and generalization of Nagel line at Dynamic Geometry Sketches Generalizes Spieker circle and associated Nagel line.