Unit I: Preface & The Anatomy of a Geometric Construction — Summary

Structural & Heuristic Overview — Forord og Indledning — Analyse, Konstruktion, Bevis og Diskussion (pp. 1–6 (with Problem 42, p. 15) · Foundations & Problem 42)

Architectural Overview (pp. 1–6)

In his six-page Preface and Introduction, Julius Petersen transforms what had for two millennia been treated as an uncodified art of geometric ingenuity into an explicit algorithmic methodology:

• Why Theorems Alone Do Not Teach Construction (pp. 1–2): Knowing deductive theorems (Euclid's Elements) does not automatically reveal which auxiliary lines or circles unlock a construction problem. Petersen organizes 410 graded problems around general heuristic operators—giving brief hints rather than full solutions so students internalize the method itself.

• Instrument Restriction & Quadratic Algebra (p. 3): Restricting geometric constructions to the straight edge (Postulate 1) and pair of compasses (Postulate 3) is announced from the outset as equivalent to solving algebraic systems solely through rational operations and iterated square roots (proved rigorously in Appendix III, pp. 97–102).

• Taxonomy of Determinateness (p. 4): - Over-determinate (more than determinate): Excess independent conditions; solvable only under special compatibility relations (theorems). - Determinate: Exactly enough conditions to fix a finite number \(N \ge 0\) of discrete solutions. - Indeterminate: One or more omitted conditions, causing unknown points or lines to sweep out continuous geometrical loci.

• The Four-Part Canonical Schema (p. 4): 1. Analysis — Assume the required figure already drawn and trace relations linking sought elements to given ones; 2. Mode of Proceeding (Construction) — Execute those relations forward from the given data by straight lines and circles; 3. Proof (Synthesis) — Verify deductively that every condition of the problem is satisfied; 4. Discussion (Diorismos) — Determine parameter inequalities separating 0, 1 (tangent), 2 (secant), or \(\infty\) (porismatic) real solutions.

• Two Master Heuristics (pp. 5–6): 1. Relaxation of One Condition (Method of Two Loci & Method of Similitude); 2. Transformation of the Figure (Parallel Translation, Replacing, Axial Reflection, and Spiral Revolution).

Symbolic Schema Matrix (Problem 42, p. 15)

Four-Stage Realization in discussion_mean_proportional

Stage	Synthetic Geometric Meaning	Exact geometor.model Realization
1. Analysis	Right \(\triangle BAC\) on hypotenuse \(BC = a\) has altitude \(AD = h\) with \(AD^2 = BD \cdot DC\).	Split into Locus e (circle on diameter \(BC\)) and Locus b (parallel at height \(h\)).
2. Proceeding	Bisect \(BC\) at \(M\), draw \((M, B)\), erect perpendicular of height \(h\), draw parallel \(L_h\), intersect.	M = (5, 0), (M, B) radius 5, L_h: y = 4 \(\implies A_1 = (2, 4), A_2 = (8, 4)\).
3. Proof	Thales (\(\angle B A_1 C = 90^\circ\)) + Euclid VI.8 (\(A_1 D_1^2 = B D_1 \cdot D_1 C\)).	Q(A1, D1) = 16 = 4 * 4 = dist(B, D1) * dist(D1, C); line_spread([B, A1], [C, A1]) = 1.
4. Discussion	Discriminant \(\Delta = (a/2)^2 - h^2\): two solutions (\(h < a/2\)), one at apex \(K = (5, 5)\) (\(h = a/2\)), zero (\(h > a/2\)).	Exact discriminant 25 - 16 = 9 > 0 verified symbolically in \(\mathbb{Q}\).