F.32 Markowski’s Theorem

Circumcircle Midpoint Sagitta Product, Euler’s Inequality, and Midpoint Inversion

If J, K, L denote the perpendicular sagittas from the side midpoints of triangle ABC to the circumcircle, Markowski’s Theorem states J K L = (1/2) r^2 R = (R^2 - d_IO^2)^2 / (8 R).

Interactive GEOMETOR construction for F.32 (Markowski’s Theorem). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point ===== O = circumcenter(A, B, C) I = incenter(A, B, C) c_out = circle(O, A) M_A = midpoint(B, C) P_A = intersection(ray(O, M_A), c_out) J = segment(M_A, P_A)

Mathematical Synthesis

Dated April 23, 2015 (SeeligNotebooks), Seelig’s two-page investigation GIVEN :math:`d_{IO}` and :math:`R` — what is :math:`Jcdot Kcdot L`? (:person:`Markowski`’s Theorem) examines the three sagitta segments \(J, K, L\) extending from the side midpoints MA, MB, MC to the circumcircle cout. Combining Markowski's theorem \(JKL = \frac{1}{2}r^2 R\) with Euler’s identity \(d_{IO} = \sqrt{R^2 - 2rR}\), Seelig eliminates the inradius \(r = (R^2 - d_{IO}^2)/(2R)\) to express the sagitta product purely in terms of \(R\) and \(d_{IO}\): \(JKL = \frac{(R^2 - d_{IO}^2)^2}{8R}\).

On Folio 1, Seelig also notes the midpoint circle-inversion radius \(\rho_I = \sqrt{R^2 - (a/2)^2}\) fixing the vertices of side \(a\). On Folio 2, he tests the formula on the equilateral triangle (\(d_{IO}=0 \implies JKL = R^3/8\)) and the \(30^\circ\text{--}60^\circ\text{--}90^\circ\) triangle (\(R=1, r=(\sqrt{3}-1)/2, d_{IO}^2 = 2-\sqrt{3}, JKL = (2-\sqrt{3})/4\)). In a candid archival moment (“Stumped! FWS 4/23/2015”), Seelig substitutes \(d_{IO}^2 = 2-\sqrt{3}\) into \((R^2 - (d_{IO}^2)^2)^2/(8R)\), accidentally squaring \(d_{IO}^2\) a second time—whose removal immediately reconciles both sides to \((2-\sqrt{3})/4\).

Key Algebraic Identities

\[J K L = \frac{1}{2} r^2 R = \frac{(R^2 - d_{IO}^2)^2}{8 R}, \qquad d_{IO}^2 = R^2 - 2 r R\]

\[J + K + L = 2 R + r, \qquad \rho_I = \sqrt{R^2 - \left(\frac{a}{2}\right)^2}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.32 Markowski’s Theorem — Folios (Folios 0221–0222, 2 scanned manuscript pages in resources/seelig/32_markowski)
Euclid’s Elements	III.35, IV.5 (EuclidElements)
GEOMETOR Elements (G-Index)	G.7 plane surface, G.8 boundary
Encyclopedia of Triangle Centers	X(1) — Incenter, X(3) — Circumcenter (Kimberling1998)
Seelig Spreadsheets	Circumcircle-Circumcenter, Incircle-Incenter (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.4 Carnot’s Theorem, F.11 Circumcenter, F.29 Inradius

Required for: F.41 Geometric Problems and Extremal Loci