F.24 Gothic Arch

Pointed Vesica Arch in a Square and Tangency Equations for the Inscribed Oculus Circle

In a square ABCD of side s with pointed Gothic arch arcs of radius s centered at A(0, 0) and B(s, 0) meeting at apex E(s/2, (sqrt(3)/2)s), an inscribed tracery circle centered at P(s/2, Q) of radius r = s - p satisfies p*s = P_x H_x + P_y H_y at its tangency point H.

Interactive GEOMETOR construction for F.24 (Gothic Arch). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point ===== ( A B ) = circle(A, B) ( B A ) = circle(B, A) C, D = intersect(( A B ), ( B A )) M = midpoint(A, B)

Mathematical Synthesis

Rooted in the vesica piscis of EuclidElements I.1 (F.18), the pointed Gothic arch in Topic 24 (SeeligNotebooks) is bounded by two circular arcs AB and BA of radius \(s\) inside a square \(ABCD\) of side \(s\), together with diagonal radials \(l_{CF}: y = \sqrt{3}x + s(1-\sqrt{3})\) and \(l_{DF}: y = -\sqrt{3}x + s\).

Seelig sets up the system of three circle equations for an oculus circle of radius \(r\) centered on the symmetry axis at \(P(s/2, Q)\), proves the tangency dot-product identity \(p s = P_x H_x + P_y H_y\) (where \(p = |AP| = s - r\)), and uses the semicircle geometric mean to show \(g^2 = |AG|^2 = 2 s G_x\).

Key Algebraic Identities

\[r^2 = \left(x - \frac{s}{2}\right)^2 + (y - Q)^2, \qquad p s = P_x H_x + P_y H_y \quad (r = s - p), \qquad g^2 = G_x^2 + G_y^2 = 2 s G_x\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.24 Gothic Arch — Folios (Folios 0135–0139, 5 scanned manuscript pages in resources/seelig/24_gothic)
Euclid’s Elements	I.1, III.11, III.12 (EuclidElements)
GEOMETOR Elements (G-Index)	G.1 point, G.3 extremities of a segment
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	General Reference (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.9 Circle Equations, F.10 Circles and Geometric Mean, F.18 Equilateral Triangle