F.42 Altitude of a Pythagorean Triple Triangle

Inverse Pythagorean Theorem, Hypotenuse Projections, and Incircle Touchpoint Distinction

In a right triangle with legs a, b and hypotenuse c, the altitude h_c = a b / c satisfies the Inverse Pythagorean Theorem 1/h_c^2 = 1/a^2 + 1/b^2 and divides the hypotenuse into projections u = b^2/c and y = a^2/c with h_c^2 = u y.

Interactive GEOMETOR construction for F.42 (Altitude of a Pythagorean Triple Triangle). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point ===== c_hyp = segment(A, B) H_C = foot(C, line(A, B)) h_c = segment(C, H_C) I = incenter(A, B, C) T_C = foot(I, line(A, B))

Mathematical Synthesis

Titled The :term:`Altitude <orthocenter>` of a :term:`Pythagorean Triple` Triangle (SeeligNotebooks), this four-folio study examines the right-triangle altitude hc dropped from the right angle C to the hypotenuse A B (Euclid Elements VI.8). Equating twice the area \(2K = ab = ch_c\) gives \(h_c = ab/c\) and the Inverse Pythagorean Theorem \(\frac{1}{h_c^2} = \frac{a^2+b^2}{a^2 b^2} = \frac{1}{a^2} + \frac{1}{b^2}\), while the hypotenuse projections \(u = b^2/c\) and \(y = a^2/c\) satisfy the geometric mean identity \(h_c = \sqrt{uy}\).

On Folio 3, Seelig highlights a classic geometric pitfall inside a bold warning box: the cevian from C to the incircle touchpoint TC (which splits \(c\) into \(s-a\) and \(s-b\) with \(K = (s-a)(s-b)\)) does not coincide with the perpendicular altitude foot HC (“CT_c is not perp to BA”) unless \(a = b\).

Key Algebraic Identities

\[h_c = \frac{a b}{c}, \qquad \frac{1}{h_c^2} = \frac{1}{a^2} + \frac{1}{b^2}, \qquad u = \frac{b^2}{c},\; y = \frac{a^2}{c},\; h_c^2 = u y\]

\[|A T_C| = s - a \neq u = \frac{b^2}{c} \quad (a \neq b), \qquad K = (s - a)(s - b) = \frac{a b}{2}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.42 Altitude of a Pythagorean Triple Triangle — Folios (Folios 0267–0270, 4 scanned manuscript pages in resources/seelig/42_pythagorean-triples)
Euclid’s Elements	I.47, VI.8 (EuclidElements)
GEOMETOR Elements (G-Index)	G.4 segment, G.11 rectilineal figures
Encyclopedia of Triangle Centers	X(1) — Incenter, X(4) — Orthocenter, X(7) — Gergonne Point (Kimberling1998)
Seelig Spreadsheets	Pythagorean Triples, Right Triangle, Altitudes (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.10 Circles and Geometric Mean, F.29 Inradius, F.33 Euclidean (m, n) Generators

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