Critical & Symbolic Notes — Unit V: All Nine Points in Feuerbach’s 9. Satz (p. 62)

Did Feuerbach Already Construct the Euler Midpoints in 1822?

A famous commonplace in secondary histories of mathematics asserts that Feuerbach noticed only six points on the nine-point circle (the three altitude feet \(M, N, P\) and the three side midpoints \(a, b, c\)) and overlooked the three Euler midpoints of \(AO, BO, CO\).

Visual inspection of page 62 (`9. Satz`, `fb_082.jpg`) refutes that oversimplification! Read lines 8–14 of 9. Satz verbatim:

1. Feuerbach joins the side midpoint \(D\) of \(AB\) to the midpoint \(L\) of \(OK\), prolonging \(DL\) until it meets the altitude \(CP\) at \(J\).

2. By congruent triangles \(\triangle OLJ \cong \triangle KLD\) [I. 26], he proves both:

   • \(OJ = KD = \frac{1}{2}CO\) — so \(J\) is literally the Euler midpoint E₃ bisecting the vertex-to-orthocenter segment \(CO\)!

   • \(JL = DL = \frac{1}{2}CK = \frac{1}{2}R\) — so \(J\) lies at exact distance \(\frac{1}{2}R\) from \(L\), diametrically opposite the side midpoint \(D\) on the circle \((L, \frac{1}{2}R)\)!

Thus, although Feuerbach states the verbal enunciation of § 56 and 9. Satz in terms of the altitude feet and side midpoints, his synthetic proof on page 62 explicitly constructs the Euler midpoint \(J = \operatorname{mid}(CO)\) as the antipode of the side midpoint \(D = \operatorname{mid}(AB)\) with \(JL = DL = LP = \frac{1}{2}R\)!

Reflection Unfolding in 8. Satz: Equally noteworthy on pp. 61–62 is 8. Satz, where prolonging \(PG, PH\) to \(J, K\) on line \(MN\) unfolds the orthic perimeter \(JK = MN + MP + NP\) by reflection across \(AC\) and \(BC\)—anticipating H. A. Schwarz’s celebrated reflection proof of Fagnano’s minimal-perimeter problem by more than half a century!