Chapter VIII: Circles — Commentary

Constructive Analysis, Historical Provenance & GEOMETOR Notes (0191–0248, pp. 172–229)

Mathematical Architecture & Circle Systems

Spanning 160 articles (§§ 358–517, pp. 172–229), Chapter VIII builds the complete synthetic theory of systems of one, two, and three circles:

  1. Inverse Points & Orthogonal Circles (§§ 358–372): Inverse points (\(OP \cdot OP' = R^2\)) and orthogonal circles (\(r_1^2 + r_2^2 = d^2\)), proving that circles through a pair of inverse points cut the reference circle orthogonally.

  2. Poles & Polars (§§ 373–393): La Hire's reciprocity theorem (§ 379), self-polar (conjugate) triangles, and harmonic chord division (pole and polar).

  3. Centers of Similitude, Power & Radical Axis (§§ 394–439): Monge's theorem on the six centers of similitude of three circles (§ 405), Steiner's power of a point (\(d^2 - R^2\)), and the radical axis of two circles (generalizing Chapter I, § 11, Locus 12).

  4. Coaxal Pencils & Casey’s Theorem (§§ 440–490): Intersecting, tangent, and non-intersecting coaxal circles, Poncelet's limiting points \(L, L'\), conjugate orthogonal pencils, and Casey's generalized Ptolemy theorem (§ 471, Sequel to Euclid).

  5. Three Circles & The Problem of Apollonius (§§ 491–517): Radical center, circles of antisimilitude, the ten classical cases (PPP through CCC) of the Problem of Apollonius, and Gergonne's celebrated master solution (§ 517) constructing complementary pairs of up to eight tangent circles via poles of axes of similitude and the radical center!