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The common incircle of the medial triangle 
(left figure) and the congruent triangle 
, where 
are the midpoints of the line
segment joining the Nagel point Na with the
vertices of the original triangle 
(right figure).
The Spieker circle has circle function
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which does not correspond to any named center. The center of the Spieker circle is called the Spieker center Sp, and the circle has radius
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where 
is the inradius and 
is the semiperimeter of
the reference triangle.
The Spieker circle passes through Kimberling centers 
for 
, 3036, 3037, 3038, 3039, 3040, 3041, and 3042.
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The common area of Wolfram Research's front-end programming group contains a triangular table illustrating the construction of the Spieker circle. This table was built by
Theodore Gray, director of user interfaces at Wolfram Research, using walnut and
inlays of maple, the latter of which was obtained from a tree formerly standing in
the yard of front end developer Chris Carlson. The triangular table 
has sides lengths (3, 4, 5), a Pythagorean
triple. The larger inlaid circle is the incircle
of 
,
with the incenter 
representing the point of concurrence of the triangle's angle bisectors. The smaller inlaid circle is the
Spieker circle, which can be seen to correspond to the incircle
of the medial triangle 
. The triangle's cleavers
are also shown, and concur in the Spieker center
(which is therefore also the cleavance center).
E. Pegg Jr. has posted a photo history of the construction of this table.
and the Spieker Circle." §1.4 in