Unit VII: Inverse Figures, Circle Systems & Constructibility Criteria — Notes

Historical, Mathematical & Cross-Repository Commentary (pp. 33–37 & Appendix pp. 89–102)

Historical, Mathematical & GEOMETOR Commentary

1. Petersen’s Malfatti Proof (1879, pp. 95–97) & Circle Inversion

In 1803 Gian Francesco Malfatti posed the problem of cutting three cylindrical columns out of a triangular marble prism (inscribing three mutually tangent circles in a triangle, each tangent to two sides). In 1826 Jakob Steiner stated without proof an astonishingly simple synthetic construction using the incircles of the three sub-triangles formed by the angle bisectors, remarking that it flowed from theorems on centres of similitude and radical axes. After Andrew Hart (1856) and Heinrich Schröter (1874) gave intricate proofs, Petersen’s 2-page elementary proof in Problem 404 (pp. 95–97) became famous throughout Europe for needing nothing beyond equal tangents and inscribed-angle chasing (Theorem 1 on circular-arc triangles).

2. Petersen’s Algebra of Constructibility & Hilbert’s Chapter VII

Petersen was equally distinguished in algebra (Theorie der algebraischen Gleichungen (1878)) and graph theory (where the Petersen graph, 1898, and Petersen’s theorem on 3-regular bridgeless graphs bear his name). His Appendix III (pp. 97–102) bridges field theory and synthetic geometry:

  • Building on Pierre Laurent Wantzel's 1837 theorem (Liouville's Journal) and Petersen’s own 1871 dissertation (Om Ligninger, der løses ved Kvadratrod), Petersen translates tower degree \([\mathbb{F} : \mathbb{Q}] = 2^k\) into curve-order criteria (Theorems 1–4, pp. 98–99), allowing students to prove geometric impossibility directly from locus degree without setting up cubic polynomials!

  • Compare Hilbert’s Grundlagen, Chapter VII (§§ 36–39, 1899) (geometor/hilbert): where Petersen characterizes full straightedge + compass via arbitrary real quadratic radical towers, Hilbert refines the criterion to straightedge + segment-transferer (Eichmass), isolating totally real Pythagorean radical towers \(\sqrt{\alpha^2 + \beta^2}\)!

3. The 1879 Pre-Lindemann Witness on \(\pi\) (p. 101)

Petersen’s remark on page 101 of our 1879 edition is a jewel of nineteenth-century mathematical history:

  • In 1873, Charles Hermite had proved the transcendence of \(e\).

  • In 1879, when Petersen penned and Haagensen translated “which, as yet, nobody has been able to prove” for \(\pi\), squaring the circle remained the single surviving unsolved classical Greek construction problem!

  • Just three years later (June 1882), Ferdinand von Lindemann (Über die Zahl π, Math. Ann. 20) proved \(e^{i\pi} + 1 = 0\) impossible for algebraic \(\pi \neq 0\), establishing that \(\pi\) is transcendental—and a fortiori outside Petersen’s quadratic radical towers.

4. Exact geometor.model Realization: tangent_circle_by_inversion

In geometor.petersen.constructions.tangent_circle_by_inversion() (Problem 199, p. 36), we construct a circle tangent to three given circles concurrent at \(O = (0, 0)\) with centers \(M_1 = (-12, 0)\), \(M_2 = (0, -9)\), \(M_3 = (12, 9)\) and inversion circle \(\Omega\) of radius \(R_0 = 12\) (\(I = 144\)):

  • Inverting the antipodal points \(A_1 = (-24, 0)\), \(A_2 = (0, -18)\), \(A_3 = (24, 18)\) across \(\Omega\) yields \(A'_1 = (-6, 0)\), \(A'_2 = (0, -8)\), \(A'_3 = (\frac{48}{5}, \frac{36}{5})\).

  • The three inverse lines \(x = -6\), \(y = -8\), and \(4x + 3y = 24\) bound the exact \((18, 24, 30)\) right triangle \(\triangle V_1 V_2 V_3\) with \(V_1 = (12, -8)\), \(V_2 = (-6, 16)\), \(V_3 = (-6, -8)\), incenter \(J' = (0, -2)\), and inradius \(r' = 6\).

  • Inverting the incircle’s three contact points \(T'_1 = (-6, -2)\), \(T'_2 = (0, -8)\), \(T'_3 = (\frac{24}{5}, \frac{8}{5})\) back across \(\Omega\) produces exact rational tangency points

    \[T_1 = \left(-\frac{108}{5},\, -\frac{36}{5}\right), \qquad T_2 = (0, -18), \qquad T_3 = (27, 9),\]

    whose circumcircle is the exact tangent circle \(\omega\) centered at \(J = (0, 9)\) of integer radius \(R = 27\) (with center distances \(|J M_1| = 15 = 27 - 12\), \(|J M_2| = 18 = 27 - 9\), and \(|J M_3| = 12 = 27 - 15\)—internally enveloping all three concurrent circles of radii \(12, 9, 15\)).