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The Incircle of the Medial Triangle. The center of the Spieker circle is called the Spieker Center.
See also Incircle, Medial Triangle, Spieker Center
References
Johnson, R. A.  Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle.  Boston, MA:
  Houghton Mifflin, pp. 226-228, 1929.