T_17 • The Spieker Circle¶
Category: Named Circles
The incircle of the medial triangle M_a M_b M_c, centered at the Spieker Center X10 with radius r/2, and the radical circle of the three excircles.
Primary Construction & Central Lines¶
Key Properties & Sequenced Operations¶
Definition (THEODOR SPIEKER (1888)):
Spieker Circle (Center X_10, Radius r/2) • Incircle of Medial Triangle M_a M_b M_c— The Spieker Circle: Centered at X10, the Spieker circle is inscribed in the medial triangle M_a M_b M_c. (Dilation of the incircle at X2 by factor −1/2 produces the Spieker circle of radius r/2.)Coordinate & Algebraic Law (RADICAL CENTER OF EXCIRCLES):
Midpoint of X_1 and X_8 on the Nagel Line • Barycentrics b+c : c+a : a+b— Perimeter Centroid: X10 = b+c : c+a : a+b is the complement of X1 and midpoint of X1 X8. (Equal tangents can be drawn from X10 to all three excircles of △ABC.)Central Line Incidences (MEDIAL INCIRCLE GEOMETRY):
Nagel Line {1, 2, 8, 10} ∩ Fuhrmann–Spieker Line {4, 9, 10}— Spieker Axes: The Nagel line {1, 2, 8, 10} and the Fuhrmann–Spieker axis {4, 9, 10} meet at X10. (Every property of X1 in △ABC translates to X10 inside the medial triangle M_a M_b M_c.)Collection 1 Network (COLLECTION 1 ANCHOR):
X_10 • X_1 • X_2 • X_8 • X_4 • X_9— Spieker Network: X10 bisects X1 and X8 through X2, and aligns with X4 and X9. (Connects the incenter, centroid, orthocenter, Nagel point, and Mittenpunkt.)