F.35 Obtuse and Acute Triangles

Euclid’s Excess and Deficit (II.12 & II.13), Altitude-Median Offsets, and Apollonius’s Theorem

By Euclid’s Elements II.12 (excess) and II.13 (deficit), the square on side a opposite an obtuse or acute angle satisfies a^2 = b^2 + c^2 +/- 2 c x, where x = |a^2 - (b^2 + c^2)| / (2 c) is the projection of side b onto line c and h = sqrt(b^2 - x^2) is the altitude.

Interactive GEOMETOR construction for F.35 (Obtuse and Acute Triangles). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point ===== l_ab = line(A, B) H_C = foot(C, l_ab) M_C = midpoint(A, B) h_c = segment(C, H_C) x_proj = segment(A, H_C) d_mh = segment(M_C, H_C)

Mathematical Synthesis

Titled OBTUSE AND ACUTE TRIANGLES / EUCLID’S CONCEPT OF ‘EXCESS’ AND ‘DEFICIT’ (SeeligNotebooks), this seven-folio study recovers Euclid’s geometric precursors to the Law of Cosines: II.12 (excess \(+2cx\) for obtuse triangles) and II.13 (deficit \(-2cx\) for acute triangles). Without invoking trigonometric functions, Seelig computes the altitude foot projection \(x = \pm\frac{a^2-(b^2+c^2)}{2c}\) and altitude \(h = \sqrt{b^2-x^2}\) directly from the three side lengths.

Across Folios 2–7, Seelig works out three canonical integer triangles in complete detail: the obtuse \((3,5,7)\) Eisenstein triangle (\(120^\circ\) angle, \(x = 1.5, h_B = \frac{3\sqrt{3}}{2}\) and \(x = \frac{65}{14}, h_c = \frac{15\sqrt{3}}{14}\)), the acute \((5,6,7)\) triangle (\(x = 1\), division ratio \(r/r' = 28/9\)), and the acute \((7,8,9)\) triangle, computing the exact distance \(|m_A h_A| = \frac{17}{14}, |m_B h_B| = \frac{33}{16}, |m_C h_C| = \frac{5}{6}\) between each midpoint and altitude foot, before closing with Apollonius’s median identity \(D^2 = \frac{U^2+V^2}{2} - (E/2)^2\).

Key Algebraic Identities

\[a^2 = b^2 + c^2 \pm 2 c x \implies x = \frac{|a^2 - (b^2 + c^2)|}{2 c}, \qquad h^2 = b^2 - x^2\]

\[|M_C H_C| = \left|\frac{c}{2} - x\right| = \frac{|a^2 - b^2|}{2 c}, \qquad m_c^2 = \frac{a^2 + b^2}{2} - \left(\frac{c}{2}\right)^2\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.35 Obtuse and Acute Triangles — Folios (Folios 0225–0231, 7 scanned manuscript pages in resources/seelig/35_obtuse-and-acute)
Euclid’s Elements	II.12, II.13 (EuclidElements)
GEOMETOR Elements (G-Index)	G.3 extremities of a segment, G.11 rectilineal figures
Encyclopedia of Triangle Centers	X(2) — Centroid, X(4) — Orthocenter (Kimberling1998)
Seelig Spreadsheets	General Reference, 5-6-7 A,B,C, Altitudes (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.3 Biquadratic Forms in Heron’s Formula, F.6 Centroid

Required for: F.36 Orthic Triangle, F.41 Geometric Problems and Extremal Loci, F.46 Stewart’s Theorem, F.47 The UVW Coordinate Triangle, F.49 Vesica Piscis and Acute-Obtuse Probability