Technical Glossary

Constructive & Minimal-Instrument Terminology (Français · Italiano · Deutsch · English)

Aehnlichkeitspunkte (A, I) (German · External and Internal Centers of Similitude (Homothetic Centers))

In Steiner’s Zweites Kapitel (§§ 12–13) and Sechste bis Achte Aufgabe (§ 18), the harmonic centers A and I on the line of centers M M_1 through which parallel radii of the fixed drawn circle M and an un-drawn virtual circle M_1 correspond under straightedge projection.

Droite antiparallèle (French · Antiparallel Line)

A transversal cutting two sides AB, AC of angle A at P, Q such that ∠AQP = ∠ABC (and ∠APQ = ∠ACB). The locus of midpoints of antiparallels to BC is the symmedian through A—the key theorem behind Lemoine’s S = 27 construction of X(6).

Opération C₁ (French · Placing a compass point on a determined point)

Setting one leg of the compass on a previously determined point (op.: C_1). Taking up the distance AB between two determined points with the two legs of the compass requires placing both points and therefore counts as 2C_1.

Opération C₂ (French · Placing a compass point at an indeterminate point on a traced locus)

Setting one leg of the compass at an arbitrary (indeterminate) point lying on an already traced straight line or circle (op.: C_2), contributing 1 to both simplicity S and exactitude E.

Opération C₃ (French · Tracing a circle or circular arc)

Describing a circle (or sufficient arc) with the compass once its center and aperture are established (op.: C_3). In any construction symbol, the coefficient m_3 of C_3 equals the exact number of circles drawn.

Notation O(AB) / O(R) (French · Lemoine Circle Notation)

Shorthand introduced by Lemoine throughout Géométrographie: O(AB) denotes the circle centered at O whose radius equals segment AB; O(A) or O(R) denotes the circle centered at O passing through point A (or having given radius R).

Conjugué harmonique (French · Harmonic Conjugate)

For three collinear points A, B, C, the unique fourth point D satisfying the cross-ratio (A, B; C, D) = -1 (CA/CB = -DA/DB); constructed by complete quadrilateral in Problem XLVI with pure straightedge in 9R_1 + 6R_2 (S = 15, zero circles).

Construction spéculative (French · Speculative Construction)

A construction optimized for brevity of verbal enunciation and deductive proof—reducing a new problem to a previously proved proposition regardless of hidden operational overhead—in contrast to a construction géométrographique executed directly on the drawing sheet.

Droite de Lemoine (French · Lemoine Line (Trilinear Polar of X(6)))

Polar of the Lemoine point K = X(6) with respect to the circumcircle—equivalently, the collinearity locus of the intersections of each side of triangle ABC with the circumcircle tangent at the opposite vertex (Problem LII, S = 19).

Équerre (Géométrographie de l’équerre) (French · Draftsman’s Set-Square / Right Triangle)

Auxiliary drafting triangle slid along a fixed straightedge to draw parallels and perpendiculars directly; first analyzed in Lemoine’s 1888 Oran note and formalized in Scientia No. 18 (1902) via elementary alignment and sliding operations.

Coefficient d’exactitude (E) (French · Coefficient of Exactitude (Preparatory Precision Cost))

Sum of all preparatory point-coincidence operations prior to tracing: E = l_1 + m_1 + m_2. Because every alignment of ruler edge or compass point introduces a small observational error, fewer preparatory coincidences yield higher graphical precision (paradoxically, a ‘smaller’ E denotes a more exact drawing).

Fester Hülfskreis (mit gegebenem Mittelpunct M) (German · Fixed Auxiliary Circle with Given Center M)

Steiner’s single permanent reference circle in the plane. Together with its center M, it supplies an involution of parallel diameters and orthogonal chords enabling pure straightedge simulation of all compass intersections; without M, projective collineations fixing the circle break Euclidean metric structure.

Géométrographie (French · Geometrography (Art and Calculus of Geometric Constructions))

Émile Lemoine’s quantitative discipline decomposing every geometric construction into elementary physical instrument operations in order to measure its operational simplicity (S), alignment exactitude (E), and count of traced lines and circles, replacing speculative textbook proofs with minimal constructive syntheses.

Géométrographie Mascheronienne (French · Mascheronian (Compass-Only) Geometrography)

Evaluation of constructions restricted exclusively to the compass (l_1 = l_2 = 0; symbol m_1 C_1 + m_2 C_2 + m_3 C_3) following Lorenzo Mascheroni’s La Geometria del Compasso (1797), where straight lines exist implicitly as ordered pairs of points.

Géométrographie Steinerienne (French · Steinerian (Straightedge-Only with Fixed Circle) Geometrography)

Evaluation of constructions restricted exclusively to the straightedge (m_1 = m_2 = m_3 = 0; symbol l_1 R_1 + l_2 R_2) given a single permanently drawn circle with known center M, following Poncelet (1822) and Jakob Steiner (1833).

Inversion circulaire (French · Circle Inversion)

Conformal involution mapping A ≠ O on ray OA to A’ with OA · OA’ = R^2; executed in Problem XXXVIII (for OA > R/2) by compass alone in 5C_1 + 3C_3 (S = 8, zero lines), forming the cornerstone of Adler’s 1890 proof of Mascheroni’s theorem.

Linie der gleichen Potenzen (Axe radicale) (German · Line of Equal Powers (Radical Axis))

Coined via Steiner’s ‘Potenz des Punctes’ (1826/1833, §§ 14–17) and Gaultier/Chasles’s ‘axe radical’: the straight locus of points having equal power d_1^2 - r_1^2 = d_2^2 - r_2^2 with respect to two circles, constructed in Lemoine Problem XLI in S = 17 and used by Steiner (Eighth Problem) to intersect two un-drawn circles by straightedge alone.

Division en moyenne et extrême raison (φ) (French · Division in Extreme and Mean Ratio (Golden Section))

Partition of segment AB at X such that AB · BX = AX^2, i.e., AB / AX = AX / XB = φ = (1 + √5)/2 (Euclid II.11, VI.30); constructed by Lemoine in Problem XXXVI using a single straight line and six circles (2R_1 + R_2 + 9C_1 + 6C_3, S = 18).

Point de Lemoine K = X(6) (Point symédian) (French · Lemoine Point / Symmedian Point X(6))

Isogonal conjugate of the centroid G = X(2), minimizing the sum of squared distances to the three sides of triangle ABC (trilinear coordinates a : b : c); constructed géométrographiquement in Problem LI in 4R_1 + 2R_2 + 13C_1 + 8C_3 (S = 27 vs. S = 54 classical).

Points de Brocard (ω, ω’) (French · First and Second Brocard Points)

Bicentric pair of interior points of triangle ABC satisfying ∠ωAB = ∠ωBC = ∠ωCA = ω and ∠ω’BA = ∠ω’CB = ∠ω’AC = ω; constructed simultaneously by Lemoine in Problem LV in S = 38 (versus S = 74 in Brocard’s classical three-circle construction).

Polaire d’un point par rapport à un cercle (French · Polar Line of a Pole with Respect to a Circle)

Projective harmonic locus conjugate to point A across any secant of circle O (perpendicular to OA through the inverse point A’); constructed by inscribed complete quadrangle in Problem XXXIX in 12R_1 + 7R_2 (S = 19, zero circles).

Opération R₁ (French · Straightedge alignment through one given point)

Placing the edge of the ruler in coincidence with one determined point (op.: R_1). Aligning the ruler simultaneously through two determined points A and B counts as 2R_1.

Opération R₂ (French · Drawing a straight line along the straightedge)

Tracing an indefinite straight line (or required segment) along the aligned edge of the ruler (op.: R_2). In any construction symbol, the coefficient l_2 of R_2 equals the exact number of straight lines drawn.

Coefficient de simplicité (S) (French · Coefficient of Simplicity)

Total number of elementary operations executed in a construction: S = l_1 + l_2 + m_1 + m_2 + m_3 (plus set-square coefficients when the équerre is admitted). Lower S signifies greater economy of execution.