F.8 Chord Intersection

Intersecting Chords Theorem, Sagitta Formula, Power of a Point, and Ptolemy’s Bound

If two chords AB and CD of a circle (O, R) intersect at a point X at distance r = |OX| from the center, then the product of their segments is constant and equals the power of X with respect to the circle: |AX|*|BX| = |CX|*|DX| = R^2 - |OX|^2.

Interactive GEOMETOR construction for F.8 (Chord Intersection). Hover any geometric label in the text or figure to highlight the corresponding element.

O : Point A : Point B : Point C : Point D : Point ===== circle = circle(O, A) AB = segment(A, B) CD = segment(C, D) P = intersect(AB, CD) AC = segment(A, C) DB = segment(D, B)

Mathematical Synthesis

In Topic 08 (SeeligNotebooks), Seelig proves the Intersecting Chords Theorem (EuclidElements, Proposition III.35) via the similarity of \(\triangle AXC \sim \triangle DXB\) (matching AC and DB across intersection P) and equates the common product to the power of a point \(R^2 - |OX|^2\).

He illustrates the theorem with two numerical problems: computing the sagitta \(x = \frac{5(3-\sqrt{5})}{2} \approx 1.91\) of a chord of length \(10\) in a circle of diameter \(15\), and solving \(x(10-x) = 4\cdot 4 \implies x = 2, 8\) where \(R^2 - |OX|^2 = 16\), noting that the inscribed quadrilateral \(ACBD\) satisfies Ptolemy’s relation \(AC\cdot BD + AD\cdot BC = AB\cdot CD = 80\).

Key Algebraic Identities

\[|AX| \cdot |BX| = |CX| \cdot |DX| = R^2 - |OX|^2, \qquad x_{\text{sagitta}} = R - \sqrt{R^2 - \left(\frac{c}{2}\right)^2}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.8 Chord Intersection — Folios (Folios 0050–0053, 4 scanned manuscript pages in resources/seelig/08_chord-intersection)
Euclid’s Elements	III.35, III.36 (EuclidElements)
GEOMETOR Elements (G-Index)	G.3 extremities of a segment
Encyclopedia of Triangle Centers	General projective / metric locus (Kimberling1998)
Seelig Spreadsheets	Chord Intersection (Spreadsheets)

Constructive Dependency Graph

Required for: F.9 Circle Equations, F.10 Circles and Geometric Mean, F.12 Coaxial Circles, F.13 Compare Triangles in Circles, F.15 Cyclic Quadrilateral, F.25 Harmonic Conjugates, F.26 Harmonic Conjugate Inversion