Chapter IV: The Quadrilateral — Commentary

Constructive Analysis, Historical Provenance & GEOMETOR Notes (0143–0158, pp. 124–139)

Mathematical Architecture (Added in the 1952 Second Edition)

Chapter IV (§§ 240–281, pp. 124–139) was newly written for the 1952 Revised and Enlarged Edition (2nd ed., 1952), organizing quadrilateral geometry into three progressive strata:

  • General Quadrilateral (§§ 240–250): Varignon’s parallelogram of side midpoints, bimedians, and Euler's theorem relating the sum of squares of the four sides to the squares of the diagonals plus four times the square of the segment joining the diagonal midpoints (§ 246).

  • Cyclic Quadrilateral (§§ 251–267): Ptolemy's first and second theorems (§ 255), the anticenter (Mathot point, § 260), and Brahmagupta's area formula and orthodiagonal median-altitude theorem (§§ 261–266).

  • Circumscriptible & Bicentric Quadrilaterals (§§ 268–281): Pitot’s theorem (\(a + c = b + d\)), Newton's line of a circumscribed quadrilateral, and Fuss / bicentric relations.