Chapter V: The Simson Line — Commentary

Constructive Analysis, Historical Provenance & GEOMETOR Notes (0159–0169, pp. 140–150)

Mathematical Architecture

Chapter V (§§ 282–305, pp. 140–150) investigates the pedal projections of a point \(P\) onto the sidelines of \(\triangle ABC\):

• Wallace–Simson Theorem (§ 283): The pedal feet of \(P\) are collinear on the Simson line if and only if \(P\) lies on the circumcircle (historically discovered by William Wallace in 1799, Simson having received traditional attribution).

• Orthocenter Bisection (§ 299): The Simson line of \(P\) bisects the segment \(PH\) joining \(P\) to the orthocenter \(H\)—at a point lying directly on the nine-point circle!

• Angle & Envelope Properties (§§ 290–305): Two circumcircle points \(P, P'\) have Simson lines intersecting at half the central angle of arc \(PP'\); diametrically opposite points have perpendicular Simson lines meeting on the nine-point circle (Steiner's deltoid envelope).

Cross-Repository GEOMETOR Links

• geometor/etc (Nine-Point Circle X(5), Steiner Deltoid)