F.38 Pedal Triangles

Trilinear Pedal Side Lengths, Euler’s Pedal Area Theorem, and Carnot’s Perpendicularity Law

The pedal triangle A’B’C’ of a point P(tau_A : tau_B : tau_C) has area Delta(P) = ((R^2 - OP^2)/(4 R^2)) Delta, and its six side-projection segments satisfy Carnot’s alternating square identity AC’^2 + BA’^2 + CB’^2 = C’B^2 + A’C^2 + B’A^2.

Interactive GEOMETOR construction for F.38 (Pedal Triangles). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point P : Point ===== P_A = foot(P, line(B, C)) P_B = foot(P, line(C, A)) P_C = foot(P, line(A, B)) s_ab = segment(P_A, P_B) s_bc = segment(P_B, P_C) s_ca = segment(P_C, P_A)

Mathematical Synthesis

In this two-folio study (SeeligNotebooks), Seelig summarizes the metric and coordinate laws of the pedal triangle \(\triangle A'B'C'\) (labeled PA, PB, PC in the construction) of a point P with homogeneous trilinear coordinates \((\tau_A : \tau_B : \tau_C)\). Folio 1 records the exact pedal side length \(a' = \frac{abc\sqrt{\tau_B^2 + \tau_C^2 + 2\tau_B\tau_C\cos\alpha}}{2R(a\tau_A + b\tau_B + c\tau_C)}\) and Euler’s pedal area theorem \((\Delta P) = \frac{R^2 - OP^2}{4R^2}\Delta\), which vanishes precisely when P lies on the circumcircle (Simson’s line theorem).

Folio 2 pairs Carnot’s alternating segment square identity \({AC'}^2 + {BA'}^2 + {CB'}^2 = {C'B}^2 + {A'C}^2 + {B'A}^2\) (annotated with Seelig’s marginal exclamation “wow!”) with the \(3\times 3\) trilinear coordinate matrix for the vertices \(A', B', C'\).

Key Algebraic Identities

\[a' = \frac{a b c \sqrt{\tau_B^2 + \tau_C^2 + 2\tau_B\tau_C\cos\alpha}}{2 R (a\tau_A + b\tau_B + c\tau_C)}, \qquad \Delta(P) = \frac{R^2 - OP^2}{4 R^2}\Delta\]

\[{AC'}^2 + {BA'}^2 + {CB'}^2 = {C'B}^2 + {A'C}^2 + {B'A}^2\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.38 Pedal Triangles — Folios (Folios 0236–0237, 2 scanned manuscript pages in resources/seelig/38_pedal-triangles)
Euclid’s Elements	I.12, I.47 (EuclidElements)
GEOMETOR Elements (G-Index)	G.4 segment, G.11 rectilineal figures
Encyclopedia of Triangle Centers	X(1) — Incenter, X(3) — Circumcenter, X(4) — Orthocenter (Kimberling1998)
Seelig Spreadsheets	Pedal Triangle, Orthic 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,	—	—	—	—	—

Constructive Dependency Graph

Prerequisites: F.4 Carnot’s Theorem, F.36 Orthic Triangle