F.50 Altitudes, Mitchell’s Reciprocal Heron Formula, and the Symphonic Theorem

The Sweet Theorem (h = ab/c), Reciprocal Altitude Area Formula, and Six Orthocenter Identities

Given the three altitudes h_A, h_B, h_C of triangle ABC with reciprocal semiperimeter H = (h_A^{-1} + h_B^{-1} + h_C^{-1})/2, Mitchell’s reciprocal Heron formula states K^{-1} = 4 sqrt(H (H - h_A^{-1})(H - h_B^{-1})(H - h_C^{-1})), and a triangle is right-angled at C if and only if h_A^{-2} + h_B^{-2} = h_C^{-2} (the Symphonic Theorem).

Interactive GEOMETOR construction for F.50 (Altitudes, Mitchell’s Reciprocal Heron Formula, and the Symphonic Theorem). 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_a = segment(A, H_A) h_b = segment(B, H_B) h_c = segment(C, H_C) H = intersection(line(A, H_A), line(B, H_B))

Mathematical Synthesis

Across four folios (SeeligNotebooks), Seelig synthesizes the metric laws of the three altitude segments ha, hb, hc. Folio 1 (Altitude of a Right Triangle (the sweet theorem)) derives \(h = ab/c\) and the Inverse Pythagorean Theorem \(h^{-2} = a^{-2} + b^{-2}\). Folios 2–3 record Mitchell’s (2005) reciprocal Heron's formula (Mitchell2005): defining \(H = \frac{1}{2}(h_A^{-1}+h_B^{-1}+h_C^{-1})\), the reciprocal area is \(K^{-1} = 4\sqrt{H(H-h_A^{-1})(H-h_B^{-1})(H-h_C^{-1})}\), which immediately recovers each side via \(a = 2K/h_A\). Seelig also names the Symphonic Theorem: since \(h_A = b, h_B = a, h_C = ab/c\) in a right triangle, the three altitudes satisfy \(h_A^{-2} + h_B^{-2} = h_C^{-2}\).

Folio 4 compiles six classical orthocenter identities in Conway triangle notation, including the sum of upper altitude segments \(t_A + t_B + t_C = |AH| + |BH| + |CH| = 2(r + R)\) and the altitude-chord product \(|AH|\cdot|HH_A| = |BH|\cdot|HH_B| = |CH|\cdot|HH_C|\).

Key Algebraic Identities

\[H = \frac{h_A^{-1} + h_B^{-1} + h_C^{-1}}{2}, \qquad \frac{1}{K} = 4\sqrt{H(H - h_A^{-1})(H - h_B^{-1})(H - h_C^{-1})}\]

\[\frac{1}{h_A^2} + \frac{1}{h_B^2} = \frac{1}{h_C^2} \quad (\gamma = 90^\circ), \qquad |AH| + |BH| + |CH| = 2(r + R)\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.50 Altitudes, Mitchell’s Reciprocal Heron Formula, and the Symphonic Theorem — Folios (Folios 0318–0321, 4 scanned manuscript pages in resources/seelig/50_altitudes)
Euclid’s Elements	I.12, I.47, VI.8 (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	Altitudes, Orthocenter, Right 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: 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.3 Biquadratic Forms in Heron’s Formula, F.14 Conway Notation, F.34 Nine-Point Circle, F.42 Altitude of a Pythagorean Triple Triangle