F.34 Nine-Point Circle

Feuerbach’s Nine-Point Circle, Euler Line Collinearity, and Orthocenter Distance Identities

The circle Gamma_N passing through the three altitude feet H_A, H_B, H_C also passes through the three side midpoints M_A, M_B, M_C and the three Euler midpoints; its radius is r_N = R/2 and its center N = X(5) bisects segment OH on the Euler line.

Interactive GEOMETOR construction for F.34 (Nine-Point Circle). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point ===== M_A = midpoint(B, C) M_B = midpoint(C, A) M_C = midpoint(A, B) O = circumcenter(A, B, C) H = orthocenter(A, B, C) N = midpoint(O, H) c_nine = circle(N, M_A)

Mathematical Synthesis

In this concentrated summary folio (SeeligNotebooks), Seelig defines the nine-point circle \(\Gamma_N\) (“The circle that passes through all three footers is the nine point circle”), recording its center N (X(5) — Nine-Point Center), radius \(r_N = \frac{1}{2}R\), and two expressions for the distance between the circumcenter O (X(3) — Circumcenter) and orthocenter H (X(4) — Orthocenter): \(|OH| = \sqrt{2r^2 + 4R^2 - S_\omega}\) (where \(S_\omega = \frac{1}{2}(a^2+b^2+c^2)\)) and \(|OH|^2 = AH^2 + BH^2 + CH^2 - 3R^2 = 9R^2 - (a^2+b^2+c^2)\).

A pencil annotation highlights that N lies on the Euler line through O, G, H at the midpoint of O H, illustrated by a compass drawing of the obtuse \((2,3,4)\) triangle showing the exterior altitude feet and their common circle \(\Gamma_N\). On the canonical \((5,6,7)\) triangle, \(R = \frac{35\sqrt{6}}{24}\), \(r_N = \frac{35\sqrt{6}}{48}\), and \(|OH|^2 = 9\left(\frac{1225}{96}\right) - 110 = \frac{155}{32}\).

Key Algebraic Identities

\[r_N = \frac{1}{2} R, \qquad N = \frac{O + H}{2} = X(5), \qquad |ON| = \frac{1}{2}|OH|\]

\[|OH|^2 = 9 R^2 - (a^2 + b^2 + c^2) = 2 r^2 + 4 R^2 - \frac{a^2 + b^2 + c^2}{2}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.34 Nine-Point Circle — Folios (Folios 0224–0224, 1 scanned manuscript page in resources/seelig/34_nine-point-circle)
Euclid’s Elements	IV.5 (EuclidElements)
GEOMETOR Elements (G-Index)	G.7 plane surface
Encyclopedia of Triangle Centers	X(2) — Centroid, X(3) — Circumcenter, X(4) — Orthocenter, X(5) — Nine-Point Center (Kimberling1998)
Seelig Spreadsheets	9-Point Circle, Euler Line, Orthocenter (Spreadsheets)

Computational Workbook Verification

The following worksheets from seelig’s Triangles and Circles.ods workbook verify the numerical and symbolic formulas for this topic on the canonical \((a, b, c) = (5, 6, 7)\) triangle:

Worksheet: Orthocenter (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
I5	I5	A	=D12
J5	A	0	=E12
K5	0	0	=F12
H6	H6	B	=D13
I6	B	6	=E13
J6	6	0	=F13
I7	I7	C	=D14
J7	C	-2	=E14

Cell Excerpt

Orthocenter	—	—	—	—	—
The orthocenter H is the intersecti	—	—	—	—	—
When the reference triangle is obtus	—	—	—	—	—
and has radius rPOLAR2 = 4R2 - (a2+b	—	—	—	—	—
Fred Seelig	—	—	—	—	—
Begun 6/1/2011 and completely rework	—	—	—	—	—

Constructive Dependency Graph

Prerequisites: F.6 Centroid, F.11 Circumcenter

Required for: F.36 Orthic Triangle, F.50 Altitudes, Mitchell’s Reciprocal Heron Formula, and the Symphonic Theorem, F.52 Homogeneous Barycentric Coordinates