Spieker circle

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 ("X"10 according to Kimberling's enumeration) is collinear with the incenter ("X"1), centroid ("X"2), and Nagel point ("X"8). Indeed, the distances between pairs of the four points have constant ratios, regardless of the shape of triangle "ABC", as indicated here:

:displaystyle |X_8X_{10}| : |X_8X_2| : |X_8X_1| = 3 : 4 : 6

References

*cite book
author = Johnson, Roger A.
title = Modern Geometry
publisher = Houghton Mifflin
location = Boston
year = 1929
Dover reprint, 1960.

*cite journal
author = Kimberling, Clark
title = Triangle centers and central triangles
journal = Congressus Numerantium
volume = 129
year = 1998
pages = i-xxv, 1–295


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