Summary — Unit V: Synthetic Euclidean Demonstrations (1.–9. Satz, pp. 56–62)

Pure Compass-and-Straightedge Architecture of the Anhang

Dependency Map of Feuerbach’s Nine Synthetic Propositions

Proposition	Synthetic Statement	Euclidean Elements Dependencies
1.–2. Satz	\(CO = 2\,KD\) and \(\overline{AB}^2 + \overline{CO}^2 = 4R^2\)	Books I, III ([III. 3], [III. 31], [I. 47])
3.–5. Satz	\(r(a+b+c) = R(MN+MP+NP)\) and \(\triangle MNP : \Delta = \rho : R\)	[VI. 16]
6. Satz	\(CO \cdot OP = 2R\rho\)	[III. 21], [III. 22], [III. 31], [IV. 4], [VI. 4], [VI. 16]
7. Satz	\(\overline{AB}^2 + \overline{AC}^2 + \overline{BC}^2 = 8R^2 + 4\rho R\)	[I. 47], [II. 3], [II. 4], [II. 5], [II. 12], [VI. 4]
8. Satz	\(GH = \tfrac{1}{2}(MN + MP + NP)\) (reflection unfolding of Fagnano’s perimeter!)	[I. 15], [I. 26], [III. 21], [III. 22], [VI. 2], [VI. 4]
9. Satz	\(L = \operatorname{mid}(OK)\) satisfies \(LM = LN = LP = LD = LJ = \tfrac{1}{2}R\)	[I. 4], [I. 15], [I. 26], [I. 29], [I. 33], [VI. 2]