F.36 Orthic Triangle

Pedal Triangle of the Orthocenter, Tangential Homothety, and Fagnano’s Minimum-Perimeter Problem

The orthic triangle H_A H_B H_C formed by the altitude feet of an acute triangle ABC is the pedal triangle of the orthocenter H = X(4), is homothetic to the tangential triangle, has H as its incenter, and solves Fagnano’s 1775 problem by minimizing the perimeter among all triangles inscribed in ABC.

Interactive GEOMETOR construction for F.36 (Orthic Triangle). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point ===== H_A = foot(A, line(B, C)) H_B = foot(B, line(C, A)) H_C = foot(C, line(A, B)) H = intersection(line(A, H_A), line(B, H_B)) s_ab = segment(H_A, H_B) s_bc = segment(H_B, H_C) s_ca = segment(H_C, H_A)

Mathematical Synthesis

On Folio orthic012.jpg (SeeligNotebooks), Seelig synthesizes the three defining properties of the orthic triangle \(\triangle H_A H_B H_C\): (1) it is the pedal triangle of the orthocenter H (X(4) — Orthocenter), whose altitudes bisect the interior angles of \(\triangle H_A H_B H_C\) so that H is the incenter of the orthic triangle; (2) its sides are anti-parallel to \(BC, CA, AB\) and parallel to the tangents to the circumcircle at A, B, C, making it homothetic to the tangential triangle; and (3) for any acute \(\triangle ABC\), it solves Fagnano’s Problem (1775), achieving the minimum perimeter \(p_{\text{orthic}} = 2K/R = 4R\sin\alpha\sin\beta\sin\gamma\) among all inscribed triangles.

On the canonical \((5,6,7)\) acute triangle (\(K = 6\sqrt{6}, R = \frac{35\sqrt{6}}{24}\)), the orthic sides are \(a\cos\alpha = \frac{19}{7}, b\cos\beta = \frac{9}{7}, c\cos\gamma = \frac{1}{5}\), giving the exact minimum Fagnano perimeter \(p_{\text{orthic}} = \frac{2K}{R} = \frac{288}{35}\).

Key Algebraic Identities

\[a_{\text{orthic}} = a \cos\alpha, \qquad b_{\text{orthic}} = b \cos\beta, \qquad c_{\text{orthic}} = c \cos\gamma\]

\[p_{\text{orthic}} = \frac{2 K}{R} = 4 R \sin\alpha\sin\beta\sin\gamma, \qquad K_{\text{orthic}} = 2 K \cos\alpha\cos\beta\cos\gamma\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.36 Orthic Triangle — Folios (Folios 0232–0232, 1 scanned manuscript page in resources/seelig/36_orthic-triangle)
Euclid’s Elements	I.12, III.32 (EuclidElements)
GEOMETOR Elements (G-Index)	G.4 segment
Encyclopedia of Triangle Centers	X(3) — Circumcenter, X(4) — Orthocenter, X(5) — Nine-Point Center (Kimberling1998)
Seelig Spreadsheets	Orthic Triangle, Orthocenter, Pedal Triangle (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: Orthic Triangle (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
G18	G18	5	=G16
H18	5	5	=H16
G19	G19	-9	=G17
H19	-9	4	=H17
G20	G20	0	=G18
H20	0	6	=H18
G22	G22	9.2	=SQRT((G18-G17)^2+(H18-H17)^2)
H22	9.2	9.2	=G23

Cell Excerpt

Orthic Triangle	—	—	—	—	—
The reference triangle DR’s orthic t	—	—	—	—	—
The three vertices are the three foo	—	—	—	—	—
The orthic center H is the incenter	—	—	—	—	—
This is an incredibly useful triangl	—	—	—	—	—
Since most sketches start with a DC,	—	—	—	—	—

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.34 Nine-Point Circle, F.35 Obtuse and Acute Triangles

Required for: F.38 Pedal Triangles