T_21 • Orthic & Tangential Triangles¶
Category: Derived Central Triangles
The Orthic Triangle H_a H_b H_c formed by the altitude feet of X4 and the Tangential Triangle formed by the tangents to the circumcircle at A, B, and C (perspector X6).
Primary Construction & Central Lines¶
Key Properties & Sequenced Operations¶
Definition (FAGNANO’S MINIMUM PERIMETER TRIANGLE (1775)):
Orthic Triangle H_a H_b H_c (Perspector X_4) • Tangential Triangle (Perspector X_6)— Orthic & Tangential Duality: The orthic triangle H_a H_b H_c of X4 is inscribed in the nine-point circle and homothetic to the tangential triangle of X6. (In any acute triangle, the orthic triangle has the minimum perimeter of all inscribed triangles.)Coordinate & Algebraic Law (ORTHIC INCENTER X4):
Parallel Corresponding Sides: H_b H_c ∥ Tangent at A (since OA ⊥ H_b H_c)— Parallel Side Theorem: Circumradii from X3 are perpendicular to the orthic sides H_a H_b H_c, making them parallel to the tangents of X6. (Altitudes AD, BE, CF bisect the interior angles of the orthic triangle H_a H_b H_c.)Central Line Incidences (CLAWSON POINT X19):
Euler Line {3, 4, 5} • Brocard Axis {3, 6} • Clawson Axis {4, 19}— Orthic–Tangential Axes: Euler line {3, 4, 5}, Brocard axis {3, 6}, and Clawson line {4, 19}. (X19 is the second center of similitude of the orthic and excentral configurations and perspector of the orthic axis.)Collection 1 Network (POLAR TRIANGLE RESPECT TO CIRCUMCIRCLE):
X_4 • X_3 • X_5 • X_6 • X_19— Orthic Network: X4, X3, X5, X6, and Clawson point X19. (The tangential triangle is the polar triangle of △ABC with respect to the circumcircle, and anticevian of X6.)