T_16 • The Nine-Point (Feuerbach) Circle

Category: Named Circles

The circle of radius R/2 centered at X₅ passing through the 3 side midpoints, 3 altitude feet, and 3 Euler midpoints of triangle ABC, tangent to the incircle at X₁₁.

Primary Construction & Central Lines

T_16 • The Nine-Point (Feuerbach) Circle Toggle Central Lines

Key Properties & Sequenced Operations

1. Definition (PONCELET, BRIANCHON & FEUERBACH (1821–1822)): Nine-Point Circle (Center X_5, Radius R/2) • 3 Midpoints + 3 Altitude Feet + 3 Euler Points — The Nine-Point Circle: Centered at X₅, the nine-point circle circumscribes both the medial triangle M_a M_b M_c and the orthic triangle. (Dilation of the circumcircle by −1/2 at X₂ or +1/2 at X₄ produces the exact same nine-point circle.)

2. Coordinate & Algebraic Law (FEUERBACH TANGENCY): Midpoint of X_3 and X_4 • Tangency with Incircle at Feuerbach Point X_11 — Dual Homothety: X₅ is the complement of X₃ and midpoint of X₃ X₄, tangent to the incircle at X₁₁. (|X₁ X₅| = |R/2 − r|, proving internal tangency of the incircle and nine-point circle at X₁₁.)

3. Central Line Incidences (NINE-POINT NEXUS): Euler Line {2, 3, 4, 5} ∩ Feuerbach Line {1, 5, 11, 12} — Intersection at X₅: The Euler line {2, 3, 4, 5} and Feuerbach line {1, 5, 11, 12} cross at X₅. (X₁₁ always lies strictly on the nine-point circle.)

4. Collection 1 Network (ORTHIC & MEDIAL CIRCUMCIRCLE): X_5 • X_2 • X_3 • X_4 • X_1 • X_11 • X_12 — Nine-Point Family: X₅ unites X₂, X₃, X₄ with X₁, X₁₁, X₁₂. (Because the medial and orthic triangles share the same circumcircle, their circumcenters coincide at X₅.)

Collection 1 Centers Involved

X₅, X₁, X₂, X₃, X₄, X₁₁, X₁₂