T_3 • Complement & Anticomplement¶
Category: Transformations & Conjugacies
Homothety centered at the Centroid X2 with ratio −1/2 (complement) and ratio −2 (anticomplement), mapping every triangle element to its medial or anticomplementary counterpart.
Primary Construction & Central Lines¶
Key Properties & Sequenced Operations¶
Definition (HOMOTHETIC SCALING):
Complement h(X_2, −1/2) • Anticomplement h(X_2, −2)— Centroid Homothety: Dilation at X2 by ratio −1/2 maps △ABC to the medial triangle M_a M_b M_c and the circumcircle to the nine-point circle. (Every center of the medial triangle is the exact complement of the corresponding center of △ABC.)Coordinate & Algebraic Law (COLLINEAR TRIPLETS):
u : v : w ↦ v+w : w+u : u+v ⟷ −u+v+w : u−v+w : u+v−w— Barycentric Shift: Complement maps u : v : w to v+w : w+u : u+v; anticomplement maps to −u+v+w : u−v+w : u+v−w. (For any point P, its anticomplement P’, P, X2, and complement P’’ form a 6 : 2 : 1 segment ratio.)Central Line Incidences (PARALLEL GEOMETRIES):
Euler Line {2, 3, 4, 5} & Nagel Line {1, 2, 8, 10}— Collection 1 Action: Centroid X2 is the intersection of the Nagel homothety axis {1, 2, 8, 10} and the Euler homothety axis {2, 3, 4, 5}. (The Euler line of △ABC and the Nagel line of △ABC are linked by a 1 : −2 dilation about X2.)Collection 1 Network (MIDPOINT THEOREM):
(X_8 → X_1 → X_10) • (X_4 → X_3 → X_5)— Homothetic Chains: On the Nagel axis X8 → X1 → X10; on the Euler axis X4 → X3 → X5. (X10 is the midpoint of X1 X8, and X5 is the midpoint of X3 X4.)