Technical Glossary¶
Bilingual Latin, German, and Modern Triangle-Geometry Terminology
Every inline :term: role links directly to its entry below.
Euler Line (Linea Euleri / Eulersche Gerade)¶
Tradition: Euler E325 (§§ 16–18, 1765/1767); Feuerbach (§§ 54–55, § 60, 1822)
Formulation: Collinear axis through
X(4),X(5),X(2),X(3)
The straight line passing through the orthocenter \(E = X(4)\), nine-point center \(L = X(5)\), centroid \(F = X(2)\), and circumcenter \(H = X(3)\) of any non-equilateral triangle, with \(EF : FH = 2 : 1\) and \(EL = LH = \frac{3}{4}EF\).
Nine-Point Circle (Kreis durch die Fußpunkte der Perpendikel)¶
Tradition: Brianchon & Poncelet (1821); Feuerbach (§§ 26, 54–56, 9. Satz, 1822); Terquem (1842)
Formulation: Circle of center
X(5)and radius \(R_9 = \frac{1}{2}R\)
The circle passing simultaneously through the three altitude feet (\(H_a, H_b, H_c\)), the three side midpoints (\(M_a, M_b, M_c\)), and the three Euler midpoints bisecting the segments from each vertex to the orthocenter (\(E_a, E_b, E_c\)).
Feuerbach Point X(11) (Innerer Berührungspunkt)¶
Tradition: Feuerbach (§ 57, 1822); Kimberling ETC
X(11)Formulation: Unique internal tangency point of the nine-point circle and the incircle
Because the distance between the nine-point center \(L = X(5)\) and the incenter \(S = X(1)\) satisfies \(\overline{LS}^2 = (\frac{1}{2}R - r)^2\), the nine-point circle and incircle touch internally at X(11) (or coincide when \(\triangle ABC\) is equilateral).
Tritangent Circles (Vier berührende Kreise)¶
Tradition: L’Huilier (1810); Feuerbach (Erster Abschnitt, §§ 1–18, 1822)
Formulation: Incircle \((S, r)\) and three excircles \((S', r'), (S'', r''), (S''', r''')\)
The four circles tangent to all three extended sidelines of \(\triangle ABC\), whose centers lie at the intersections of the internal and external angle bisectors and satisfy \(rr'r''r''' = \Delta^2\) and \(r' + r'' + r''' - r = 4R\).
Orthic Triangle (Fußpunktdreieck \(\triangle MNP\))¶
Tradition: Fagnano (1775); Feuerbach (Zweiter Abschnitt, §§ 19–26, 1822)
Formulation: Pedal triangle of the orthocenter
X(4)
The triangle formed by the feet of the three altitudes of \(\triangle ABC\). In an acute triangle, its perimeter is \(\frac{2\Delta}{R}\), its circumradius is \(\frac{1}{2}R\), its incenter is the orthocenter \(O = X(4)\), and its inradius is \(\rho = 2R\cos\alpha\cos\beta\cos\gamma\).
Feuerbach Triangle (\(\triangle Fe_a Fe_b Fe_c\))¶
Tradition: Feuerbach (§ 57, 1822); Modern Triangle Geometry
Formulation: Triangle of external nine-point / excircle tangency points
The triangle inscribed in the nine-point circle whose vertices are the three points where the nine-point circle touches the excircles externally (\(LS^{(k)} = \frac{1}{2}R + r^{(k)}\)).
Quadrance¶
Tradition: Euler (§ 18, 1765); Feuerbach (§§ 49–60, 1822); Wildberger (2005)
Formulation: Exact squared distance \(Q(P, Q) = (x_Q - x_P)^2 + (y_Q - y_P)^2\)
Polynomial metric invariant over \(\mathbb{Q}(a, b, c)\) free of square-root branch ambiguities.
Harmonic Range (\((X_4, X_2; X_5, X_3) = -1\))¶
Tradition: Pappus (Book VII); Projective Geometry
Formulation: Collinear quadruple with cross-ratio \(-1\)
Along the Euler line, the nine-point center X(5) and circumcenter X(3) divide the segment from the orthocenter X(4) to the centroid X(2) internally and externally in ratio \(3 : 1\).
Casus Irreducibilis¶
Tradition: Cardano (1545); Euler (E325 § 34, p. 123)
Formulation: Three-real-root cubic regime requiring trigonometric trisection
In Euler’s inverse triangle reconstruction \(z^3 - pz^2 + qz - r = 0\), whenever three distinct positive side lengths \((a, b, c)\) exist, the cubic discriminant is strictly positive, placing radical extraction in the casus irreducibilis.
Elementary Symmetric Polynomials \((p, q, r)\)¶
Tradition: Euler (E325 § 10, p. 107)
Formulation: \(p = a + b + c\), \(q = ab + ac + bc\), \(r = abc\)
The three algebraic generators of symmetric functions of the triangle sides, in terms of which Heron’s formula reads \(16A^2 = -p^4 + 4p^2q - 8pr\).