T_7 • The Nagel Line¶
Category: Named Central Lines
The central line passing through the Incenter X1, Centroid X2, Nagel Point X8, and Spieker Center X10, forming the tritangent counterpart to the Euler line.
Primary Construction & Central Lines¶
Key Properties & Sequenced Operations¶
Definition (CHRISTIAN HEINRICH VON NAGEL (1836)):
Nagel Line L(1, 2, 8, 10) • |X_1 X_2| : |X_2 X_10| : |X_10 X_8| = 2 : 1 : 3— The Nagel Line: Connects X1, X2, X8, and X10 across the medial triangle M_a M_b M_c and incircle. (The Spieker circle of the medial triangle is the −1/2 homothetic image of the incircle at X2.)Coordinate & Algebraic Law (PERIMETER CENTER X10):
Homothety at X_2: X_8 ↦ X_1 ↦ X_10— Barycentric Progression: X1 = a:b:c, X2 = 1:1:1, X8 = s−a:s−b:s−c, and X10 = b+c:c+a:a+b. (X10 is the center of mass of the uniform wire perimeter of triangle ABC.)Central Line Incidences (2 : 1 : 3 SEGMENT RATIOS):
Collinearity {1, 2, 8, 10}— Nagel Axis: The central line {1, 2, 8, 10} intersects the Euler line at the Centroid X2. (X10 is the exact midpoint of segment X1 X8, and X2 trisects segment X1 X8.)Collection 1 Network (COLLECTION 1 QUARTET):
X_1 • X_2 • X_10 • X_8— Ordered Nagel Sequence: X1, X2, X10, and X8 mirror the Euler quartet under inscribed geometry. (Links four fundamental centers of Collection 1 along a single homothetic axis.)