T_14 • The Incircle & Excircles¶
Category: Named Circles
The four tritangent circles tangent to all three sidelines of triangle ABC: the interior Incircle centered at X1 (radius r = Δ/s) and the three exterior Excircles.
Primary Construction & Central Lines¶
Key Properties & Sequenced Operations¶
Definition (EUCLID BOOK IV • PROPOSITION 4):
Incircle (Center X_1, Radius r = Δ/s) • Excircles (Radii r_a, r_b, r_c)— The Incircle & Excircles: Centered at X1, the incircle touches the sidelines at the intouch triangle T_a T_b T_c. (1/r = 1/r_a + 1/r_b + 1/r_c and Area(△ABC) = √(r · r_a · r_b · r_c).)Coordinate & Algebraic Law (SEVEN COLLECTION 1 CENTERS):
Touch-Point Distances: AT_b = AT_c = s−a • BT_c = BT_a = s−b • CT_a = CT_b = s−c— Tritangent Metrics: Inradius r = Δ/s, exradii r_a = Δ/(s−a), and touch segments s−a : s−b : s−c. (Generates X1 (Incenter), X7 (Gergonne), X8 (Nagel), X9 (Mittenpunkt), X10 (Spieker), X11, X12.)Central Line Incidences (RADICAL & HOMOTHETIC AXES):
Tritangent Axes {1, 7} • {1, 8} • {1, 11, 12}— Tritangent Pencils: X1 anchors the Soddy axis {1, 7}, Nagel axis {1, 8}, and Feuerbach axis {1, 11, 12}. (The Spieker center X10 is the radical center of the three excircles.)Collection 1 Network (TRITANGENT HARMONY):
X_1 • X_7 • X_8 • X_9 • X_11 • X_12— Tritangent Family: X1, X7, X8, X9, X11, and X12. (Replacing (a, b, c) with (−a, b, c) transforms the incenter into the A-excenter.)