Chapter X: Recent Geometry of the Triangle — Commentary

Constructive Analysis, Historical Provenance & GEOMETOR Notes (0263–0313, pp. 244–[294])

Mathematical Architecture & Late-19th-Century Triangle Renaissance

Spanning 153 articles (§§ 547–699, pp. 244–294), Chapter X synthesizes the explosion of triangle geometry inaugurated in the 1870s–1880s by Émile Lemoine, Henri Brocard, Joseph Neuberg, Maurice d'Ocagne, and Robert Tucker:

  1. Trilinear Poles & Polars (§§ 547–557): Projective duality with respect to a reference triangle \(ABC\), harmonically pairing inscribed Ceva configurations with circumscribed Menelaus transversals (e.g., the Lemoine axis as trilinear polar of the Lemoine point).

  2. Symmedians & Lemoine Geometry (§§ 558–596): Reflections of medians across angle bisectors (d'Ocagne), concurrency at the Lemoine (symmedian) point \(K = X(6)\) (Lemoine 1873, Grebe), least-squares distance property (\(\sum d_i^2\) minimum, § 581), and the first and second (cosine) Lemoine circles (§§ 588–596).

  3. Apollonian Circles & Isodynamic Points (§§ 597–624): The three triangle Apollonian circles sharing a common radical axis (the Brocard axis \(OK\)) and intersecting in the two isodynamic points \(W, W' = X(15), X(16)\) whose pedal triangles are equilateral (§ 606).

  4. Isogonal Conjugates (§§ 625–648): Isogonal conjugation pairing \((O, H)\), \((G, K)\), and circumcircle points with lines at infinity; Miquel/pedal circle theorem.

  5. Brocard Geometry, Tucker Circles & Orthopole (§§ 649–699): Direct and retrograde Brocard points \(\Omega, \Omega'\) with Brocard angle \(\cot\omega = \cot A + \cot B + \cot C\) (Brocard 1881, Crelle), the seven-point Brocard circle on diameter \(OK\), Steiner and Tarry points \(X(99), X(98)\), Tucker & Taylor circles (§§ 684–689), and Neuberg's orthopole (§§ 690–699, Gallatly 1913).