T_23 • Incentral & Excentral Triangles

Category: Derived Central Triangles

The Incentral Triangle I_a I_b I_c (cevian triangle of X1) formed by the interior angle-bisector feet and the Excentral Triangle J_a J_b J_c (anticevian triangle of X1) formed by the three excenters.

Primary Construction & Central Lines

T_23 • Incentral & Excentral Triangles
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) [ A I_a ]: Internal Angle Bisector of ∠A [ B I_b ]: Internal Angle Bisector of ∠B [ C I_c ]: Internal Angle Bisector of ∠C incircle: Incircle of △ABC circumcircle: Circumcircle of △ABC { 1, 6 }: Incenter–Symmedian Line { 1, 3 }: OI Line (Incenter–Circumcenter Line) { 4, 9 }: Orthocenter–Spieker Line A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) I_a = Angle Bisector Foot I_a on BC (0.7912, 0.3230) I_b = Angle Bisector Foot I_b on CA (0.3571, 0.3499) I_c = Angle Bisector Foot I_c on AB (0.5455, 0.0000) X3 = CIRCUMCENTER (0.5000, 0.1021) X4 = ORTHOCENTER (0.6122, 0.3958) X6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X9 = MITTENPUNKT (0.5110, 0.1615) X1 = INCENTER (0.5714, 0.2333) A B C Ia Ib Ic X3 X4 X6 X9 X1

Key Properties & Sequenced Operations

  1. Definition (ORTHOCENTRIC SYSTEM {X1, J_a, J_b, J_c}): Incentral Triangle I_a I_b I_c (Cevian of X_1) • Excentral Triangle J_a J_b J_c (Anticevian of X_1) — Bisector Triangles: Interior angle bisectors through X1 form the incentral feet I_a, I_b, I_c, while the incircle and circumcircle govern the excentral duality. (X1 and the three excenters form an orthocentric system: each point is the orthocenter of the other three.)

  2. Coordinate & Algebraic Law (BEVAN CIRCLE (RADIUS 2R)): △ABC is the Orthic Triangle of the Excentral Triangle • Mittenpunkt X_9 is its Symmedian Point — Excentral Shift: X1 is the orthocenter of the excentral triangle, the circumcircle is its nine-point circle, and X9 is its symmedian point. (Since △ABC is the orthic triangle of J_a J_b J_c, the circumcircle of J_a J_b J_c (the Bevan circle) has radius 2R.)

  3. Central Line Incidences (MITTENPUNKT CONCURRENCE): Incenter–Symmedian–Mittenpunkt Line {1, 6, 9} • OI Line {1, 3} • Fuhrmann Axis {4, 9} — Excentral Axes: Symmedian–Mittenpunkt line {1, 6, 9}, OI line {1, 3}, and orthocenter–Mittenpunkt axis {4, 9}. (Lines from the excenters to the side midpoints M_a, M_b, M_c concur at Mittenpunkt X9.)

  4. Collection 1 Network (EXCENTRAL ↔ ORTHIC ISOMORPHISM): X_1 • X_9 • X_6 • X_3 • X_4 — Excentral Network: X1 and X9 link X6, X3, and X4. (Every theorem about the orthic triangle of △ABC has a dual theorem about △ABC inside the excentral triangle.)

Collection 1 Centers Involved

X1, X3, X4, X6, X9