Dual Minimal-Instrument Concordance

Géométrographie (1902) · Euclid’s Elements · Mascheroni Compass-Only (1797) · Poncelet–Steiner Fixed-Circle (1833)

This concordance synthesizes Émile Lemoine’s quantitative Géométrographie (lemoine-1902) with Euclid’s axiomatic canon (euclid-heath), Lorenzo Mascheroni’s compass-only constructions (mascheroni-1797, fazzari-1901), and Jakob Steiner’s straightedge-only constructions with a single fixed circle and its center (steiner-1833). Every operation vector satisfies Lemoine’s exact identities:

\[\mathrm{Op.} : (l_1 R_1 + l_2 R_2 + m_1 C_1 + m_2 C_2 + m_3 C_3), \qquad S = l_1 + l_2 + m_1 + m_2 + m_3, \qquad E = l_1 + m_1 + m_2.\]

Master Operation-Count Summary

Classical vs. Géométrographique Simplicity & Exactitude

Prob.	Benchmark Construction	Euclid / ETC	Classical Symbol (\(S, E\))	Lemoine Symbol (\(S, E\))	\(\Delta S\)
III	Straight Line Joining Two Given Points	I.Post.1	\(2R_1 + R_2\) (\(S=3, E=2\))	\(2R_1 + R_2\) (\(S=3, E=2\))	0
VIII	Laying Off a Given Segment Along a Traced Line	I.2, I.3	\(4R_1 + 2R_2 + 5C_1 + 5C_3\) (\(S=16, E=9\))	\(2C_1 + C_2 + C_3\) (\(S=4, E=3\))	+12
XI	Constructing at a Given Point an Angle Equal to a Given Angle	I.23	\(4R_1 + 2R_2 + 8C_1 + C_2 + 4C_3\) (\(S=19, E=13\))	\(2R_1 + R_2 + 5C_1 + 3C_3\) (\(S=11, E=7\))	+8
XVI	Triangle from Three Given Sides	I.22	\(6R_1 + 3R_2 + 9C_1 + C_2 + 3C_3\) (\(S=22, E=16\))	\(4R_1 + 3R_2 + 8C_1 + C_2 + 3C_3\) (\(S=19, E=13\))	+3
XVII	Parallel Through an External Point to a Given Line	I.31	\(2R_1 + R_2 + 5C_1 + 3C_3\) (\(S=11, E=7\))	\(2R_1 + R_2 + 3C_1 + 3C_3\) (\(S=9, E=5\))	+2
XVIII	Perpendicular Bisector and Midpoint of a Segment	I.10	\(2R_1 + R_2 + 3C_1 + 2C_3\) (\(S=8, E=5\))	\(2R_1 + R_2 + 2C_1 + 2C_3\) (\(S=7, E=4\))	+1
XVIII bis	Symmetric Reflection of a Point Across a Traced Line	I.12 (cor.)	\(4R_1 + 2R_2 + 5C_1 + C_2 + 3C_3\) (\(S=15, E=10\))	\(2C_1 + 2C_2 + 2C_3\) (\(S=6, E=4\))	+9
XX.1	Perpendicular to a Traced Line from an External Point	I.12	\(2R_1 + R_2 + 3C_1 + C_2 + 3C_3\) (\(S=10, E=6\))	\(2R_1 + R_2 + 3C_1 + C_2 + 2C_3\) (\(S=9, E=6\))	+1
XX.2	Perpendicular Erected at a Given Point on a Traced Line	I.11, III.31	\(2R_1 + R_2 + 3C_1 + 3C_3\) (\(S=9, E=5\))	\(4R_1 + 2R_2 + C_1 + C_3\) (\(S=8, E=5\))	+1
XXII	Bisecting a Given Rectilinear Angle	I.9	\(2R_1 + R_2 + 4C_1 + 3C_3\) (\(S=10, E=6\))	\(2R_1 + R_2 + 3C_1 + 3C_3\) (\(S=9, E=5\))	+1
XXI	Circumcircle and Circumcenter X(3) Through Three Given Points	IV.5 · X(3) Circumcenter	\(4R_1 + 2R_2 + 6C_1 + 5C_3\) (\(S=17, E=10\))	\(4R_1 + 2R_2 + 5C_1 + 4C_3\) (\(S=15, E=9\))	+2
XXVI	Tangents to a Given Circle from an External Point	III.17	\(8R_1 + 4R_2 + 4C_1 + 3C_3\) (\(S=19, E=12\))	\(5R_1 + 3R_2 + 6C_1 + 4C_3\) (\(S=18, E=11\))	+1
XXVII	Incircle and Incenter X(1) of a Given Triangle	IV.4 · X(1) Incenter	\(6R_1 + 3R_2 + 11C_1 + 10C_3\) (\(S=30, E=17\))	\(4R_1 + 2R_2 + 11C_1 + 6C_3\) (\(S=23, E=15\))	+7
XXXIII	Fourth Proportional to Three Given Segments	VI.12	\(4R_1 + 3R_2 + 11C_1 + C_2 + 6C_3\) (\(S=25, E=16\))	\(2R_1 + 2R_2 + 11C_1 + C_2 + 6C_3\) (\(S=22, E=14\))	+3
XXXV	Mean Proportional (Geometric Mean) of Two Segments	II.14, VI.13	\(4R_1 + 3R_2 + 10C_1 + C_2 + 6C_3\) (\(S=24, E=15\))	\(2R_1 + 2R_2 + 6C_1 + C_2 + 3C_3\) (\(S=14, E=9\))	+10
XXXVI	Division in Extreme and Mean Ratio (Golden Section phi)	II.11, VI.30	\(6R_1 + 3R_2 + 9C_1 + 7C_3\) (\(S=25, E=15\))	\(2R_1 + R_2 + 9C_1 + 6C_3\) (\(S=18, E=11\))	+7
XXXVIII	Inverse of a Given Point with Respect to a Given Circle	III.36, VI.8	\(4R_1 + 2R_2 + 4C_1 + 2C_3\) (\(S=12, E=8\))	\(5C_1 + 3C_3\) (\(S=8, E=5\))	+4
XXXIX	Polar Line of a Given Point with Respect to a Traced Circle	III.17, III.36	\(10R_1 + 5R_2 + 6C_1 + 4C_3\) (\(S=25, E=16\))	\(12R_1 + 7R_2\) (\(S=19, E=12\))	+6
XLI	Radical Axis of Two Non-Intersecting Traced Circles	III.35, III.36	\(18R_1 + 9R_2 + 19C_1 + 12C_3\) (\(S=58, E=37\))	\(10R_1 + 5R_2 + 2C_3\) (\(S=17, E=10\))	+41
XLVI	Harmonic Conjugate of a Point with Respect to a Segment	VI.3 (harmonic angle bisector)	\(10R_1 + 6R_2 + 4C_1 + 2C_3\) (\(S=22, E=14\))	\(9R_1 + 6R_2\) (\(S=15, E=9\))	+7
L	Centroid G = X(2) of a Given Triangle	I.10, I.34 · X(2) Centroid	\(8R_1 + 4R_2 + 3C_1 + 3C_3\) (\(S=18, E=11\))	\(6R_1 + 3R_2 + 6C_1 + 2C_3\) (\(S=17, E=12\))	+1
LI	Lemoine Symmedian Point K = X(6) of a Given Triangle	VI.4, III.21–22 · X(6) Symmedian / Lemoine Point	\(14R_1 + 7R_2 + 21C_1 + 12C_3\) (\(S=54, E=35\))	\(4R_1 + 2R_2 + 13C_1 + 8C_3\) (\(S=27, E=17\))	+27
LIV–LV	First and Second Brocard Points omega and omega’ of a Triangle	III.21, III.32 · Bicentric Brocard pair (Ω, Ω’)	\(22R_1 + 11R_2 + 24C_1 + 17C_3\) (\(S=74, E=46\))	\(12R_1 + 6R_2 + 12C_1 + 8C_3\) (\(S=38, E=24\))	+36
LXI	Nagel Point X(8) of a Given Triangle	IV.4 (excircle touchpoints) · X(8) Nagel Point	\(6R_1 + 3R_2 + 15C_1 + 10C_3\) (\(S=34, E=21\))	\(4R_1 + 2R_2 + 7C_1 + 3C_3\) (\(S=16, E=11\))	+18
Masch. §§ 66, 112	Mascheroni Compass-Only Midpoint & Intersection of Two Lines	I.Post.3, I.10	\(2R_1 + R_2 + 2C_1 + 2C_3\) (\(S=7, E=4\))	\(10C_1 + 7C_3\) (\(S=17, E=10\))	-10
Steiner § 18 Aufg. 1–8	Poncelet–Steiner Straightedge-Only Parallel & Virtual Circle Intersections	I.Post.1–2, I.31	\(2R_1 + R_2 + 3C_1 + 3C_3\) (\(S=9, E=5\))	\(11R_1 + 6R_2\) (\(S=17, E=11\))	-8

Unit I: Historical Genesis, Methodological Criteria & The Five Elementary Operations

Companion codex unit: Unit I: Historical Genesis, Methodological Criteria & The Five Elementary Operations.

III — Straight Line Joining Two Given Points

Tracer la droite joignant deux points donnés A et B

Lemoine Folio: pp. 4 | Euclid: I.Post.1 | Elements DAG: G.1 (Postulate 1) | Projective: Projective incidence join AB

• Classical / Speculative Construction: \(2R_1 + R_2\) — Simplicity \(S = 3\), Exactitude \(E = 2\) (1 lines, 0 circles).

• Lemoine Géométrographique Synthesis: \(2R_1 + R_2\) — Simplicity \(S = 3\), Exactitude \(E = 2\) (1 lines, 0 circles).

• Mascheroni Compass-Only Counterpart (Libro VIII, § 113; Libro VII, §§ 110–112 (pp. 60–63)): A straight line cannot be continuously traced without a straightedge; Mascheroni represents line AB implicitly by its point-pair (A, B) and constructs collinear multiples (§ 64) or intersections (§ 112) by compass arcs alone.

• Poncelet–Steiner Fixed-Circle Counterpart (Einleitung § 1; Drittes Kapitel § 18, II ([68], p. 50)): Primitive straightedge operation (2R_1 + R_2): joining two given points with the ruler is immediately permitted without consulting the fixed circle M.

• Critical Note: Establishes the fundamental ruler cost 2R_1 + R_2 (S = 3, E = 2) against which all straightedge alignments are measured.

VIII — Laying Off a Given Segment Along a Traced Line

Porter sur une droite tracée une longueur donnée AB

Lemoine Folio: pp. 4–5 | Euclid: I.2, I.3 | Elements DAG: G.3, G.4 (I.2, I.3) | Projective: Metric gauge / Hilbert § 36 Streckenübertrager

• Classical / Speculative Construction: \(4R_1 + 2R_2 + 5C_1 + 5C_3\) — Simplicity \(S = 16\), Exactitude \(E = 9\) (2 lines, 5 circles).

• Lemoine Géométrographique Synthesis: \(2C_1 + C_2 + C_3\) — Simplicity \(S = 4\), Exactitude \(E = 3\) (0 lines, 1 circles).

• Mascheroni Compass-Only Counterpart (Libro I, §§ 10–11 (pp. 12–13); Libro IV, §§ 72–74 (pp. 40–42)): Modern divider-compass transfer (3C_1 + C_3 from a given origin M, or 5-circle reflection/parallelogram when transferring onto an un-drawn line defined by two points).

• Poncelet–Steiner Fixed-Circle Counterpart (Drittes Kapitel, § 18, Sechste Aufgabe ([76]–[78], pp. 56–57)): Pure straightedge transfer of segment M_1 a_1 to a given origin M_2 by constructing the centers of similitude A_1, I_1 of the fixed circle M and the virtual circle (M_2, M_1 a_1).

• Critical Note: Euclid I.2–I.3 avoids assuming a divider compass that retains its opening when lifted (S = 16); Lemoine’s modern compass takes AB in 2C_1 and lays it off in C_2 + C_3 (or C_1 + C_3 from a fixed point, S = 4, E = 3).

Unit II: Fundamental Speculative vs. Géométrographique Constructions (Euclid Book I Benchmarks)

Companion codex unit: Unit II: Fundamental Speculative vs. Géométrographique Constructions (Euclid Book I Benchmarks).

XI — Constructing at a Given Point an Angle Equal to a Given Angle

Construire en un point donné d’une droite un angle égal à un angle donné

Lemoine Folio: pp. 6 | Euclid: I.23 | Elements DAG: G.24 (I.23)

• Classical / Speculative Construction: \(4R_1 + 2R_2 + 8C_1 + C_2 + 4C_3\) — Simplicity \(S = 19\), Exactitude \(E = 13\) (2 lines, 4 circles).

• Lemoine Géométrographique Synthesis: \(2R_1 + R_2 + 5C_1 + 3C_3\) — Simplicity \(S = 11\), Exactitude \(E = 7\) (1 lines, 3 circles).

• Mascheroni Compass-Only Counterpart (Libro VIII, § 114 (pp. 63–64, Fig. 51)): Given angle ABC by three points and target ray ba, constructs fourth proportionals (§ 93) or equal-radius chord transfer (5C_1 + 3C_3 when radii AB = ab are chosen equal) to locate c.

• Poncelet–Steiner Fixed-Circle Counterpart (Drittes Kapitel, § 18, Vierte Aufgabe ([75], p. 55, Fig. 21)): Draws diameters ab, cd of the fixed circle M parallel to the sides of angle AmC and ef parallel to EF, transfers the central angle via parallel chords ce and ag, and draws pq parallel to diameter gMh.

• Critical Note: Euclid’s reduction of I.23 to I.22 joins a speculative auxiliary chord across the given angle; equal-radius arc transfer cuts S from 19 to 11.

XVI — Triangle from Three Given Sides

Construire un triangle connaissant les trois côtés a, b, c

Lemoine Folio: pp. 7–8 | Euclid: I.22 | Elements DAG: G.23 (I.22)

• Classical / Speculative Construction: \(6R_1 + 3R_2 + 9C_1 + C_2 + 3C_3\) — Simplicity \(S = 22\), Exactitude \(E = 16\) (3 lines, 3 circles).

• Lemoine Géométrographique Synthesis: \(4R_1 + 3R_2 + 8C_1 + C_2 + 3C_3\) — Simplicity \(S = 19\), Exactitude \(E = 13\) (3 lines, 3 circles).

• Mascheroni Compass-Only Counterpart (Libro I, § 11 (pp. 12–13); Libro VIII, § 115 (p. 64)): Once base endpoints B, C are separated by a, two compass arcs of radii b and c (6C_1 + 2C_3) determine the third vertex A directly without tracing the three side lines.

• Poncelet–Steiner Fixed-Circle Counterpart (Drittes Kapitel, § 18, Sechste & Achte Aufgabe ([76]–[85], pp. 56–62)): Transfers lengths a, b, c via Sixth Problem and intersects the two virtual circles (B, c) and (C, b) via their radical axis (Eighth Problem) using straightedge alone.

• Critical Note: Drawing the initial carrier line as an arbitrary line R_2 (rather than joining two pre-placed points) saves 2R_1 + C_1 over textbook execution.

XVII — Parallel Through an External Point to a Given Line

Par un point donné A hors d’une droite BC, mener une parallèle à BC

Lemoine Folio: pp. 8–9 | Euclid: I.31 | Elements DAG: G.32 (I.31) | Projective: Harmonic point at infinity

• Classical / Speculative Construction: \(2R_1 + R_2 + 5C_1 + 3C_3\) — Simplicity \(S = 11\), Exactitude \(E = 7\) (1 lines, 3 circles).

• Lemoine Géométrographique Synthesis: \(2R_1 + R_2 + 3C_1 + 3C_3\) — Simplicity \(S = 9\), Exactitude \(E = 5\) (1 lines, 3 circles).

• Mascheroni Compass-Only Counterpart (Libro IV, § 80 (p. 43, Fig. 29)): Opposite-side parallelogram: with A, B on the given line and external point C, draw B(CA) and C(BA) meeting at D (5C_1 + 2C_3, S = 7, E = 5 for point D alone!).

• Poncelet–Steiner Fixed-Circle Counterpart (Drittes Kapitel, § 18, Erste Aufgabe ([68]–[71], pp. 50–52, Fig. 19)): Pure straightedge parallel via a diameter aMb of the fixed circle (providing midpoint M of ab) and complete-quadrangle harmonic construction (11R_1 + 6R_2 when line passes through M).

• Critical Note: Gaston Tarry discovered two S = 9 constructions: 1° two equal-radius circles passing through A (2R_1 + R_2 + 4C_1 + 2C_3, E = 6, 2 circles), and 2° single-aperture rhombus (2R_1 + R_2 + 3C_1 + 3C_3, E = 5, 3 circles).

XVIII — Perpendicular Bisector and Midpoint of a Segment

Tracer la perpendiculaire au milieu de AB, ou trouver le milieu de AB

Lemoine Folio: pp. 9–10 | Euclid: I.10 | Elements DAG: G.11 (I.10) | Projective: Harmonic conjugate of point at infinity

• Classical / Speculative Construction: \(2R_1 + R_2 + 3C_1 + 2C_3\) — Simplicity \(S = 8\), Exactitude \(E = 5\) (1 lines, 2 circles).

• Lemoine Géométrographique Synthesis: \(2R_1 + R_2 + 2C_1 + 2C_3\) — Simplicity \(S = 7\), Exactitude \(E = 4\) (1 lines, 2 circles).

• Mascheroni Compass-Only Counterpart (Libro III, § 66 Sol. I–V (pp. 32–33, Figs. 14–17); Libro IV, § 79 (p. 43)): Bisecting AB without a drawn line takes 7 circles in Sol. V (10C_1 + 7C_3, S = 17): double AB to E via 4 unit-radius arcs, draw E(EB) cutting B(BA) at P, p, then P(PB) and p(PB) meet at midpoint M.

• Poncelet–Steiner Fixed-Circle Counterpart (Drittes Kapitel, § 18, Zweite & Dritte Aufgabe ([71]–[75], pp. 52–55)): Draws a line parallel to AB (First Problem) to bisect AB harmonically via a complete quadrangle (Second Problem, b), then erects the perpendicular via Third Problem.

• Critical Note: Keeping an arbitrary compass opening > AB/2 avoids setting the compass on B to measure AB, saving 1 C_1 (S = 7 vs. Euclid’s equilateral-triangle S = 8).

XVIII bis — Symmetric Reflection of a Point Across a Traced Line

Construire le symétrique A’ d’un point A par rapport à une droite tracée BC

Lemoine Folio: pp. 10 | Euclid: I.12 (cor.) | Elements DAG: G.13 (reflection lemma) | Projective: Orthogonal involution

• Classical / Speculative Construction: \(4R_1 + 2R_2 + 5C_1 + C_2 + 3C_3\) — Simplicity \(S = 15\), Exactitude \(E = 10\) (2 lines, 3 circles).

• Lemoine Géométrographique Synthesis: \(2C_1 + 2C_2 + 2C_3\) — Simplicity \(S = 6\), Exactitude \(E = 4\) (0 lines, 2 circles).

• Mascheroni Compass-Only Counterpart (Libro I, § 14 (p. 14); Libro IV, § 78 (p. 43, Fig. 28)): Identical two-circle compass-only reflection! When B and C are given points on the mirror line, B(BA) and C(CA) intersect at A’ in 4C_1 + 2C_3 (or 3C_1 + 3C_3 if radii share opening), S = 6.

• Poncelet–Steiner Fixed-Circle Counterpart (Drittes Kapitel, § 18, Dritte & Sechste Aufgabe ([71]–[78], pp. 52–57)): Drops the perpendicular from A to BC meeting BC at foot M (Third Problem), then doubles AM harmonically to A’ via a parallel line (Second Problem).

• Critical Note: Crucial géométrographique primitive: Lemoine uses compass-only reflection (0 lines, 2 circles, S = 6) inside his symmedian X(6) and tangent constructions.

XX.1 — Perpendicular to a Traced Line from an External Point

Mener par un point extérieur A la perpendiculaire à une droite tracée BC

Lemoine Folio: pp. 11 | Euclid: I.12 | Elements DAG: G.13 (I.12)

• Classical / Speculative Construction: \(2R_1 + R_2 + 3C_1 + C_2 + 3C_3\) — Simplicity \(S = 10\), Exactitude \(E = 6\) (1 lines, 3 circles).

• Lemoine Géométrographique Synthesis: \(2R_1 + R_2 + 3C_1 + C_2 + 2C_3\) — Simplicity \(S = 9\), Exactitude \(E = 6\) (1 lines, 2 circles).

• Mascheroni Compass-Only Counterpart (Libro IV, § 78 (p. 43, Fig. 28)): Reflects D across AB to E in 2 arcs B(BD), A(AD) (determining the perpendicular line DE in S = 6!), and if foot M is required, bisects DE by § 66.

• Poncelet–Steiner Fixed-Circle Counterpart (Drittes Kapitel, § 18, Dritte Aufgabe A & B ([71]–[75], pp. 52–55, Fig. 20)): When BC is a diameter ab of the fixed circle, joins Pa, Pb meeting the circle at c, d; lines ad and bc meet at orthocenter p_1, and Pp_1 is perpendicular to ab (10R_1 + 5R_2, S = 15!).

• Critical Note: Instead of drawing a circle centered at A cutting BC twice and then bisecting the chord (3 circles, S = 10), Lemoine reflects A across two centres on BC (2 circles, S = 9).

XX.2 — Perpendicular Erected at a Given Point on a Traced Line

Élever en un point donné A d’une droite BC la perpendiculaire à BC

Lemoine Folio: pp. 11–12 | Euclid: I.11, III.31 | Elements DAG: G.12 (I.11), G.105 (III.31)

• Classical / Speculative Construction: \(2R_1 + R_2 + 3C_1 + 3C_3\) — Simplicity \(S = 9\), Exactitude \(E = 5\) (1 lines, 3 circles).

• Lemoine Géométrographique Synthesis: \(4R_1 + 2R_2 + C_1 + C_3\) — Simplicity \(S = 8\), Exactitude \(E = 5\) (2 lines, 1 circles).

• Mascheroni Compass-Only Counterpart (Libro IV, §§ 76–77 (pp. 42–43, Figs. 26–27)): 3-circle Thales semicircle (§ 77): with arbitrary radius r, draw A(r) and B(r) meeting at C, then trace semicircle C(r) stepping two chords from B to D; AD is perpendicular to AB (3C_1 + 5C_3 or 4C_1 + 3C_3 + stepping).

• Poncelet–Steiner Fixed-Circle Counterpart (Drittes Kapitel, § 18, Dritte Aufgabe A ([71]–[72], pp. 52–53)): Constructs orthogonal chords in the fixed circle via diameters (Euclid III.31) and draws the parallel through A on BC.

• Critical Note: Thales single-circle construction: place compass at an arbitrary point O off BC, draw O(OA) (C_1 + C_3) cutting BC again at B, draw diameter BO cutting O(OA) at D (2R_1 + R_2), and join AD (2R_1 + R_2) — only 1 circle and S = 8!

XXII — Bisecting a Given Rectilinear Angle

Diviser un angle donné BAC en deux parties égales

Lemoine Folio: pp. 13 | Euclid: I.9 | Elements DAG: G.10 (I.9)

• Classical / Speculative Construction: \(2R_1 + R_2 + 4C_1 + 3C_3\) — Simplicity \(S = 10\), Exactitude \(E = 6\) (1 lines, 3 circles).

• Lemoine Géométrographique Synthesis: \(2R_1 + R_2 + 3C_1 + 3C_3\) — Simplicity \(S = 9\), Exactitude \(E = 5\) (1 lines, 3 circles).

• Mascheroni Compass-Only Counterpart (Libro II, § 60 (p. 29); Libro VIII, § 118 (p. 65)): Bisects the subtended arc BD of circle A(AB) via Mascheroni’s arc-bisection parallelogram (§ 60) using compass arcs alone.

• Poncelet–Steiner Fixed-Circle Counterpart (Drittes Kapitel, § 18, Fünfte Aufgabe, Fall a ([75]–[76], pp. 55–56, Fig. 21)): Draws diameters ab, cd of fixed circle M parallel to sides mA, mC so central angle aMc = AmC; the line through vertex m parallel to chord ad (or cb) bisects angle AmC!

• Critical Note: Retaining the same sufficient radius R for the initial arc at A and the two intersecting arcs from B and D avoids resetting the compass, reducing S from 10 to 9.

Unit III: Circles, Tangency, Golden Section, Projective Polarity & Modern Triangle Centers

Companion codex unit: Unit III: Circles, Tangency, Golden Section, Projective Polarity & Modern Triangle Centers.

XXI — Circumcircle and Circumcenter X(3) Through Three Given Points

Circonscrire un cercle à un triangle donné ABC, ou faire passer un cercle par A, B, C

Lemoine Folio: pp. 12–13 | Euclid: IV.5 | Elements DAG: G.117 (IV.5) | ETC: X(3) Circumcenter

• Classical / Speculative Construction: \(4R_1 + 2R_2 + 6C_1 + 5C_3\) — Simplicity \(S = 17\), Exactitude \(E = 10\) (2 lines, 5 circles).

• Lemoine Géométrographique Synthesis: \(4R_1 + 2R_2 + 5C_1 + 4C_3\) — Simplicity \(S = 15\), Exactitude \(E = 9\) (2 lines, 4 circles).

• Mascheroni Compass-Only Counterpart (Libro X, §§ 143–144 (pp. 82–84, Fig. 71)): Paired with Napoleon’s Problem (§ 143, center of a drawn circle in 6 arcs): § 144 finds circumcenter O of three non-collinear points M, A, B by compass inversion/third-proportional arcs alone.

• Poncelet–Steiner Fixed-Circle Counterpart (Drittes Kapitel, § 18, Zweite & Dritte Aufgabe ([71]–[75], pp. 52–55)): Constructs the perpendicular bisectors of AB and AC by straightedge via fixed circle M; their intersection is X(3) (while the circle itself remains represented by center O and radius OA).

• Critical Note: Sharing circle B(R) across both perpendicular bisectors AB and BC cuts the circle count from 5 to 4 (3 mediatrix circles + circumcircle O(OA), S = 15).

XXVI — Tangents to a Given Circle from an External Point

Mener d’un point extérieur A les tangentes à un cercle donné O

Lemoine Folio: pp. 15–16 | Euclid: III.17 | Elements DAG: G.91 (III.17) | Projective: Pole-polar contact chord

• Classical / Speculative Construction: \(8R_1 + 4R_2 + 4C_1 + 3C_3\) — Simplicity \(S = 19\), Exactitude \(E = 12\) (4 lines, 3 circles).

• Lemoine Géométrographique Synthesis: \(5R_1 + 3R_2 + 6C_1 + 4C_3\) — Simplicity \(S = 18\), Exactitude \(E = 11\) (3 lines, 4 circles).

• Mascheroni Compass-Only Counterpart (Libro XI, § 161 (pp. 92–93)): Finds midpoint M of OA (§ 66) or uses concentric radius 2R arc and circle A(OA) to determine contact points T, T’ on O(R) by compass alone.

• Poncelet–Steiner Fixed-Circle Counterpart (Erstes Kapitel, § 10 ([26]–[30], pp. 20–23)): Pure straightedge construction! Draw two arbitrary secants from A cutting the drawn circle O, construct the polar of A via the complete inscribed quadrangle (Lemoine XXXIX, 12R_1 + 7R_2), and join A to its two intersections T, T’ with circle O.

• Critical Note: Highlights instrument duality: when circle O is already drawn, Steiner’s projective polar method needs ZERO circles (16R_1 + 9R_2 for both tangents), whereas Lemoine optimizes the metric ruler-and-compass construction to S = 18.

XXVII — Incircle and Incenter X(1) of a Given Triangle

Inscrire un cercle dans un triangle donné ABC

Lemoine Folio: pp. 16–18 | Euclid: IV.4 | Elements DAG: G.116 (IV.4) | ETC: X(1) Incenter

• Classical / Speculative Construction: \(6R_1 + 3R_2 + 11C_1 + 10C_3\) — Simplicity \(S = 30\), Exactitude \(E = 17\) (3 lines, 10 circles).

• Lemoine Géométrographique Synthesis: \(4R_1 + 2R_2 + 11C_1 + 6C_3\) — Simplicity \(S = 23\), Exactitude \(E = 15\) (2 lines, 6 circles).

• Mascheroni Compass-Only Counterpart (Libro X, § 149 (p. 85)): Determines incenter X(1) and inradius by compass-only angle bisections (§ 118) and line intersection (§ 112) + reflection (§ 78).

• Poncelet–Steiner Fixed-Circle Counterpart (Drittes Kapitel, § 18, Dritte & Fünfte Aufgabe ([71]–[76], pp. 52–56)): Bisects angles B and C via parallel diameters of fixed circle M (Fifth Problem) to meet at incenter I = X(1), then drops perpendicular from I to BC (Third Problem) to obtain touchpoint D.

• Critical Note: One of Lemoine’s signature triumphs: eliminates 4 circles and 1 line (S: 30 -> 23) by reusing a single compass radius across vertices and measuring the perpendicular distance from I to BC without tracing the altitude line!

XXXIII — Fourth Proportional to Three Given Segments

Construire la quatrième proportionnelle x = bc/a à trois longueurs a, b, c

Lemoine Folio: pp. 24–25 | Euclid: VI.12 | Elements DAG: G.167 (VI.12) | Projective: Thales intercept / Hilbert § 15 segment product

• Classical / Speculative Construction: \(4R_1 + 3R_2 + 11C_1 + C_2 + 6C_3\) — Simplicity \(S = 25\), Exactitude \(E = 16\) (3 lines, 6 circles).

• Lemoine Géométrographique Synthesis: \(2R_1 + 2R_2 + 11C_1 + C_2 + 6C_3\) — Simplicity \(S = 22\), Exactitude \(E = 14\) (2 lines, 6 circles).

• Mascheroni Compass-Only Counterpart (Libro V, §§ 93–95 (pp. 48–49)): Remarkably compact concentric-circle construction! Draw concentric circles O(a) and O(b); place chord BC = c in O(a), then step equal arcs BD = CE on O(b) from B and C to read chord DE = bc/a directly in 4 circles!

• Poncelet–Steiner Fixed-Circle Counterpart (Drittes Kapitel, § 18, Erste & Sechste Aufgabe ([68]–[78], pp. 50–57)): Transfers segments onto two intersecting rays from O (Sixth Problem) and draws the Thales parallel via fixed circle M (First Problem).

• Critical Note: Fazzari’s 1901 Palermo preface (p. xi) singles out Mascheroni’s concentric-circle fourth proportional (§ 93, S = 14 when a <= 2c!) as simpler even than standard ruler-and-compass Thales constructions.

XXXV — Mean Proportional (Geometric Mean) of Two Segments

Construire la moyenne proportionnelle entre deux longueurs données a et b

Lemoine Folio: pp. 26–27 | Euclid: II.14, VI.13 | Elements DAG: G.63 (II.14), G.168 (VI.13) | Projective: Pappus Book III three-means configuration

• Classical / Speculative Construction: \(4R_1 + 3R_2 + 10C_1 + C_2 + 6C_3\) — Simplicity \(S = 24\), Exactitude \(E = 15\) (3 lines, 6 circles).

• Lemoine Géométrographique Synthesis: \(2R_1 + 2R_2 + 6C_1 + C_2 + 3C_3\) — Simplicity \(S = 14\), Exactitude \(E = 9\) (2 lines, 3 circles).

• Mascheroni Compass-Only Counterpart (Libro VI, §§ 99–103 (pp. 51–54)): Extracts sqrt(ab) by compass arcs alone using power-of-a-point / semicircle chord relations without drawing the diameter line.

• Poncelet–Steiner Fixed-Circle Counterpart (Drittes Kapitel, § 18, Dritte & Siebente Aufgabe ([71]–[82], pp. 52–60)): Intersects the altitude at the junction of a and b with the virtual semicircle on diameter a + b via Seventh Problem (line-circle intersection with fixed circle M).

• Critical Note: The Lemoine–Gouzy construction cuts S by 42% (24 -> 14), halving the circle count from 6 to 3.

XXXVI — Division in Extreme and Mean Ratio (Golden Section phi)

Diviser une droite donnée AB en moyenne et extrême raison

Lemoine Folio: pp. 27–28 | Euclid: II.11, VI.30 | Elements DAG: G.60 (II.11), G.185 (VI.30) | Projective: Q(phi) harmonic/golden range

• Classical / Speculative Construction: \(6R_1 + 3R_2 + 9C_1 + 7C_3\) — Simplicity \(S = 25\), Exactitude \(E = 15\) (3 lines, 7 circles).

• Lemoine Géométrographique Synthesis: \(2R_1 + R_2 + 9C_1 + 6C_3\) — Simplicity \(S = 18\), Exactitude \(E = 11\) (1 lines, 6 circles).

• Mascheroni Compass-Only Counterpart (Libro II, §§ 38–42 (pp. 22–24); Libro IX, § 128 (p. 70)): Mascheroni’s 5-circle regular decagon/pentagon division: stepping three hexagon chords B, C, D of radius r = AB around O(r) gives AC = BD = r*sqrt(3); intersecting A(AC) and D(BD) at E gives OE = r*sqrt(2), whence the decagon chord AB/phi and extreme-and-mean section (§ 128) follow by compass alone.

• Poncelet–Steiner Fixed-Circle Counterpart (Drittes Kapitel, § 18, Dritte, Sechste & Siebente Aufgabe ([71]–[82], pp. 52–60)): Erects perpendicular BC = AB/2 at B by straightedge (Second & Third Problems) and intersects line AC with virtual circle C(CB) via Seventh Problem, then transfers segment to AB via Sixth Problem.

• Critical Note: Lemoine traces ONLY ONE straight line (the perpendicular at B obtained simultaneously with the midpoint arc) instead of three lines in classical II.11/VI.30, cutting S from 25 to 18.

XXXVIII — Inverse of a Given Point with Respect to a Given Circle

Construire l’inverse A’ d’un point A par rapport à un cercle O(R)

Lemoine Folio: pp. 29–30 | Euclid: III.36, VI.8 | Elements DAG: G.110 (III.36) | Projective: Inversive duality

• Classical / Speculative Construction: \(4R_1 + 2R_2 + 4C_1 + 2C_3\) — Simplicity \(S = 12\), Exactitude \(E = 8\) (2 lines, 2 circles).

• Lemoine Géométrographique Synthesis: \(5C_1 + 3C_3\) — Simplicity \(S = 8\), Exactitude \(E = 5\) (0 lines, 3 circles).

• Mascheroni Compass-Only Counterpart (Libro V, § 86 (p. 46, Fig. 35); August Adler (1890) inversion proof of Mohr–Mascheroni): Identical 3-circle compass-only inversion (when OA > R/2): draw A(AO) cutting O(R) at M, N; then M(MO) and N(NO) intersect at A’ with OA * OA’ = R^2! (When OA <= R/2, double OA k times first.)

• Poncelet–Steiner Fixed-Circle Counterpart (Erstes Kapitel, § 10 ([26]–[30], pp. 20–23)): Pure straightedge inversion! Construct the polar p of A with respect to the fixed circle O via an inscribed quadrangle, join OA, and intersect OA with p at A’.

• Critical Note: August Adler (1890) proved the entire Mohr–Mascheroni theorem at a stroke by observing that Lemoine’s 5C_1 + 3C_3 inversion maps lines and circles into circles and lines using the compass alone.

XXXIX — Polar Line of a Given Point with Respect to a Traced Circle

Construire la polaire d’un point donné A par rapport à un cercle tracé O

Lemoine Folio: pp. 30–31 | Euclid: III.17, III.36 | Elements DAG: G.91 (III.17) | Projective: Pappus Book VII harmonic polarity

• Classical / Speculative Construction: \(10R_1 + 5R_2 + 6C_1 + 4C_3\) — Simplicity \(S = 25\), Exactitude \(E = 16\) (5 lines, 4 circles).

• Lemoine Géométrographique Synthesis: \(12R_1 + 7R_2\) — Simplicity \(S = 19\), Exactitude \(E = 12\) (7 lines, 0 circles).

• Mascheroni Compass-Only Counterpart (Libro V, § 86; Libro IV, § 77 (pp. 42–46)): Constructs the inverse A’ of A in circle O (5C_1 + 3C_3) and a second point on the perpendicular to OA at A’ (§ 77).

• Poncelet–Steiner Fixed-Circle Counterpart (Erstes Kapitel, § 10, III–IV ([27]–[29], pp. 21–22)): Exact same pure-straightedge inscribed quadrangle construction (12R_1 + 7R_2, S = 19, 0 circles), which works whether A lies outside or inside the circle and without needing the center O!

• Critical Note: Demonstrates that even inside standard ruler-and-compass Géométrographie, pure projective straightedge synthesis beats metric compass-and-ruler constructions (S = 19 vs. S = 25).

XLI — Radical Axis of Two Non-Intersecting Traced Circles

Construire l’axe radical de deux cercles tracés non sécants

Lemoine Folio: pp. 31–32, 57–58 | Euclid: III.35, III.36 | Elements DAG: G.109, G.110 (III.35–36) | Projective: Coaxal pencil of circles

• Classical / Speculative Construction: \(18R_1 + 9R_2 + 19C_1 + 12C_3\) — Simplicity \(S = 58\), Exactitude \(E = 37\) (9 lines, 12 circles).

• Lemoine Géométrographique Synthesis: \(10R_1 + 5R_2 + 2C_3\) — Simplicity \(S = 17\), Exactitude \(E = 10\) (5 lines, 2 circles).

• Mascheroni Compass-Only Counterpart (Libro VI–VII, §§ 103, 112 (pp. 54, 61–62)): Divides line of centers OO’ at M where OM^2 - O’M^2 = R^2 - R’^2 via compass-only fourth proportional and erects a perpendicular point (§ 77).

• Poncelet–Steiner Fixed-Circle Counterpart (Zweites Kapitel, §§ 15–17 ([61]–[67], pp. 45–50); Achte Aufgabe ([82]–[85], pp. 60–62)): Steiner coins the term ‘Linie der gleichen Potenzen’ and constructs the radical axis of two arbitrary circles by straightedge alone via their centers of similitude with the fixed circle M!

• Critical Note: In his famous reply to Amédée Mannheim (pp. 57–58), Lemoine contrasts Mannheim’s midpoint-of-common-tangents construction (S = 58!) against two arbitrary secant circles C_3 + C_3 yielding radical center and axis in S = 17.

XLVI — Harmonic Conjugate of a Point with Respect to a Segment

Construire le conjugué harmonique D d’un point C par rapport à A et B

Lemoine Folio: pp. 34–35 | Euclid: VI.3 (harmonic angle bisector) | Elements DAG: G.158 (VI.3) | Projective: Pappus Book VII Prop. 131 complete quadrilateral

• Classical / Speculative Construction: \(10R_1 + 6R_2 + 4C_1 + 2C_3\) — Simplicity \(S = 22\), Exactitude \(E = 14\) (6 lines, 2 circles).

• Lemoine Géométrographique Synthesis: \(9R_1 + 6R_2\) — Simplicity \(S = 15\), Exactitude \(E = 9\) (6 lines, 0 circles).

• Mascheroni Compass-Only Counterpart (Libro V, §§ 93, 96 (pp. 48–49)): Computes BD from AD : BD = AC : BC via proportional segment division (§ 96) with compass arcs alone.

• Poncelet–Steiner Fixed-Circle Counterpart (Erstes Kapitel, §§ 4–6 ([10]–[16], pp. 9–13)): Identical pure-straightedge complete quadrilateral construction (9R_1 + 6R_2, S = 15, 0 circles) — didn’t even require the fixed circle M!

• Critical Note: Projective invariant cross-ratio (A, B; C, D) = -1 constructed with zero compass operations; choosing the first auxiliary ray as R_1 + R_2 through A saves 3R_1 over naive alignment.

L — Centroid G = X(2) of a Given Triangle

Construire le centre de gravité G d’un triangle donné ABC

Lemoine Folio: pp. 42–43 | Euclid: I.10, I.34 | Elements DAG: G.11 (I.10), G.35 (I.34) | ETC: X(2) Centroid | Projective: Pappus centroidal median concurrency

• Classical / Speculative Construction: \(8R_1 + 4R_2 + 3C_1 + 3C_3\) — Simplicity \(S = 18\), Exactitude \(E = 11\) (4 lines, 3 circles).

• Lemoine Géométrographique Synthesis: \(6R_1 + 3R_2 + 6C_1 + 2C_3\) — Simplicity \(S = 17\), Exactitude \(E = 12\) (3 lines, 2 circles).

• Mascheroni Compass-Only Counterpart (Libro X, § 150 (p. 85); Libro III, § 68 (p. 36)): Finds midpoint M_a of BC (§ 66) and trisects median AM_a at G via compass-only trisection (§ 68).

• Poncelet–Steiner Fixed-Circle Counterpart (Drittes Kapitel, § 18, Erste & Zweite Aufgabe ([68]–[71], pp. 50–52)): Draws parallels to two sides via fixed circle M to bisect them harmonically, then intersects two medians by straightedge.

• Critical Note: Lemoine forms parallelogram ABDC with only 2 circles B(AC) and C(AB) so AD bisects BC without drawing BC’s perpendicular bisector, cutting S from 18 to 17 (2 circles instead of 3).

LI — Lemoine Symmedian Point K = X(6) of a Given Triangle

Construire le point de Lemoine K d’un triangle donné ABC

Lemoine Folio: pp. 43–45 | Euclid: VI.4, III.21–22 | Elements DAG: G.159 (VI.4) | ETC: X(6) Symmedian / Lemoine Point | Projective: Harmonic pole of Lemoine line / Gergonne-symmedian duality

• Classical / Speculative Construction: \(14R_1 + 7R_2 + 21C_1 + 12C_3\) — Simplicity \(S = 54\), Exactitude \(E = 35\) (7 lines, 12 circles).

• Lemoine Géométrographique Synthesis: \(4R_1 + 2R_2 + 13C_1 + 8C_3\) — Simplicity \(S = 27\), Exactitude \(E = 17\) (2 lines, 8 circles).

• Mascheroni Compass-Only Counterpart (Libro IV, § 78; Libro VII, § 112 (pp. 43, 61–62)): Lemoine’s antiparallel-reflection construction is already nearly compass-pure (8 circles, only 2 symmedian lines)! Intersecting those two symmedian point-pairs via Mascheroni § 112 yields X(6) by compass alone.

• Poncelet–Steiner Fixed-Circle Counterpart (Erstes Kapitel, § 10; Drittes Kapitel, § 18 ([26]–[30], [68]–[75])): Once circumcircle (ABC) is drawn, X(6) is constructed by PURE STRAIGHTEDGE without a single compass operation: construct the tangents at A, B, C via polars and join each vertex to the pole of its opposite side!

• Critical Note: Lemoine discovered X(6) in 1873 (AFAS Lyon, ‘Sur quelques propriétés d’un point remarquable du triangle’). In Problem LI he halves the trilinear-coordinate cost (S: 54 -> 27) by reflecting antiparallel midpoints using only 2 traced lines!

LIV–LV — First and Second Brocard Points omega and omega’ of a Triangle

Construire les points de Brocard ω et ω’ d’un triangle donné ABC

Lemoine Folio: pp. 46–48, 58–59 | Euclid: III.21, III.32 | Elements DAG: G.95 (III.21), G.106 (III.32) | ETC: Bicentric Brocard pair (Ω, Ω’) | Projective: Harmonic/isogonal Brocard circle on diameter OK

• Classical / Speculative Construction: \(22R_1 + 11R_2 + 24C_1 + 17C_3\) — Simplicity \(S = 74\), Exactitude \(E = 46\) (11 lines, 17 circles).

• Lemoine Géométrographique Synthesis: \(12R_1 + 6R_2 + 12C_1 + 8C_3\) — Simplicity \(S = 38\), Exactitude \(E = 24\) (6 lines, 8 circles).

• Mascheroni Compass-Only Counterpart (Libro VII, §§ 110–112; Libro X, § 144 (pp. 60–63, 83–84)): Constructs the centers of the adjoint tangent circles by compass-only perpendiculars and circumcenter arcs, locating ω and ω’ as second intersections of the adjoint circles.

• Poncelet–Steiner Fixed-Circle Counterpart (Drittes Kapitel, § 18, Erste, Dritte & Achte Aufgabe ([68]–[85], pp. 50–62)): Constructs parallels and perpendiculars at A, B, C via fixed circle M to locate the adjoint circle centers, then intersects the virtual adjoint circles via their radical axes (Eighth Problem).

• Critical Note: In his reply to Henri Brocard (pp. 58–59), Lemoine dissects Brocard’s own classical three-adjoint-circle trace (S = 74) and cuts the simultaneous cost nearly in half (S = 38; single Brocard point LIV takes 8R_1 + 4R_2 + 10C_1 + 6C_3, S = 28).

LXI — Nagel Point X(8) of a Given Triangle

Construire le point de Nagel X(8) d’un triangle donné ABC

Lemoine Folio: pp. 50 | Euclid: IV.4 (excircle touchpoints) | Elements DAG: G.116 (IV.4) | ETC: X(8) Nagel Point | Projective: Isotomic/cevian concurrency

• Classical / Speculative Construction: \(6R_1 + 3R_2 + 15C_1 + 10C_3\) — Simplicity \(S = 34\), Exactitude \(E = 21\) (3 lines, 10 circles).

• Lemoine Géométrographique Synthesis: \(4R_1 + 2R_2 + 7C_1 + 3C_3\) — Simplicity \(S = 16\), Exactitude \(E = 11\) (2 lines, 3 circles).

• Mascheroni Compass-Only Counterpart (Libro I, §§ 10–11; Libro VII, § 112 (pp. 12–13, 61–62)): Transfers side lengths along the extended sides by three compass arcs to mark two excircle touchpoints T_b, T_c, then intersects cevians AT_b and BT_c by § 112.

• Poncelet–Steiner Fixed-Circle Counterpart (Drittes Kapitel, § 18, Sechste Aufgabe ([76]–[78], pp. 56–57)): Transfers side lengths along the triangle’s sides via centers of similitude with fixed circle M to locate T_b, T_c, then joins AT_b and BT_c by straightedge.

• Critical Note: Instead of constructing two excircles via exterior angle bisectors (S = 34), Lemoine lays off semiperimeter differences directly with 3 circles and joins two Nagel cevians in S = 16!

Unit IV: Restricted & Alternative Instruments — Équerre, Mascheroni Compass-Only & Poncelet–Steiner Fixed-Circle

Companion codex unit: Unit IV: Restricted & Alternative Instruments — Équerre, Mascheroni Compass-Only & Poncelet–Steiner Fixed-Circle.

Masch. §§ 66, 112 — Mascheroni Compass-Only Midpoint & Intersection of Two Lines

Géométrographie Mascheronienne : milieu de AB et intersection de deux droites au compas seul

Lemoine Folio: pp. Scientia 1902; Mascheroni pp. 33, 61–63 | Euclid: I.Post.3, I.10 | Elements DAG: G.2 (Postulate 3), G.11 (I.10) | Projective: Mohr–Mascheroni / Adler inversive completeness

• Classical / Speculative Construction: \(2R_1 + R_2 + 2C_1 + 2C_3\) — Simplicity \(S = 7\), Exactitude \(E = 4\) (1 lines, 2 circles).

• Lemoine Géométrographique Synthesis: \(10C_1 + 7C_3\) — Simplicity \(S = 17\), Exactitude \(E = 10\) (0 lines, 7 circles).

• Mascheroni Compass-Only Counterpart (Libro III, § 66 Sol. V (p. 33, Fig. 14); Libro VII, §§ 110–112 (pp. 60–63, Figs. 46–50)): Proves compass-only completeness: line-circle intersection (§§ 110–111) and line-line intersection (§ 112, via symmetric points c, d across AB, parallelogram CDd-delta, and fourth proportional c-delta : CD = Cc : CS) reduce all Euclidean constructions to C_1, C_2, C_3.

• Poncelet–Steiner Fixed-Circle Counterpart (Einleitung § 2 ([2]–[4], pp. 4–5)): Steiner opens his 1833 treatise by contrasting Mascheroni’s exclusion of the straightedge with the dual projective program of excluding the compass altogether.

• Critical Note: In Géométrographie Mascheronienne (r1 = r2 = 0), bisecting an un-drawn segment AB takes 10C_1 + 7C_3 (S = 17, 7 circles), whereas reflecting a point takes 4C_1 + 2C_3 (S = 6) and constructing a parallel point takes 5C_1 + 2C_3 (S = 7).

Steiner § 18 Aufg. 1–8 — Poncelet–Steiner Straightedge-Only Parallel & Virtual Circle Intersections

Géométrographie Steinerienne : parallèle et intersections de cercles à la règle seule avec un cercle fixe

Lemoine Folio: pp. Scientia 1902; Steiner [68]–[85] (pp. 50–62) | Euclid: I.Post.1–2, I.31 | Elements DAG: G.1 (Postulate 1), G.32 (I.31) | Projective: Poncelet–Steiner Theorem (complete quadrangle + centers of similitude + radical axis)

• Classical / Speculative Construction: \(2R_1 + R_2 + 3C_1 + 3C_3\) — Simplicity \(S = 9\), Exactitude \(E = 5\) (1 lines, 3 circles).

• Lemoine Géométrographique Synthesis: \(11R_1 + 6R_2\) — Simplicity \(S = 17\), Exactitude \(E = 11\) (6 lines, 0 circles).

• Mascheroni Compass-Only Counterpart (Libro VII, §§ 110–112 (pp. 60–63)): Dual counterpart: where Mascheroni simulates straight lines via pairs of points and fourth-proportional arcs, Steiner simulates un-drawn circles via centers of similitude and radical axes.

• Poncelet–Steiner Fixed-Circle Counterpart (Drittes Kapitel, § 18, Erste bis Achte Aufgabe ([68]–[85], pp. 50–62, Figs. 19–24)): Proves straightedge-only completeness given ONE fixed circle with center M: parallel to a diameter aMb takes 11R_1 + 6R_2 (S = 17, 6 lines, 0 circles); Seventh & Eighth Problems find line-circle and circle-circle intersections of un-drawn circles via Aehnlichkeitspunkte A, I and the radical axis.

• Critical Note: Notice that center M IS indispensable: by Poncelet–Steiner–Hilbert projective invariance, a projective transformation fixing a circle without fixing its center destroys Euclidean parallelism and orthogonality!