F.19 Equal Incircles Theorem

Japanese Sangaku Ray-Fan Theorem Proved via Hyperbolic Tangents

If a fan of rays from apex P to a baseline cuts triangle ABC into consecutive subtriangles whose incircles have equal radii r_1 = r_2 = … = r_n = r, then the ray angles gamma_n with the altitude PR (of height h) follow the hyperbolic progression tan(gamma_n) = sinh(a + n*b) with tanh(b/2) = r/(h - r), so overlapping pairs of subtriangles also have equal incircles.

Interactive GEOMETOR construction for F.19 (Equal Incircles Theorem). Hover any geometric label in the text or figure to highlight the corresponding element.

A : Point B : Point C : Point ===== I = incenter(A, B, C) T_C = foot(I, line(A, B)) incircle = circle(I, T_C)

Mathematical Synthesis

The Equal Incircles Theorem is a celebrated Japanese Sangaku theorem: if rays from apex C to base A B partition \(\triangle ABC\) into subtriangles with equal inradii \(r\), then the incircles of the triangles formed by every second ray (or every \(k\)-th ray) are also equal.

In Topic 19 (SeeligNotebooks), Seelig gives an elegant proof (marked ‘interesting!!!’): using similar triangles \(\triangle OWX \sim \triangle PQX\) at height \(h - r\), he derives \((h - r)(\tan\gamma_{n+1} - \tan\gamma_n) = r(\sec\gamma_n + \sec\gamma_{n+1})\) and shows that the hyperbolic parameterization \(\tan\gamma_n = \sinh(a + nb), \sec\gamma_n = \cosh(a + nb)\) satisfies this recurrence identically with constant step \(\tanh(b/2) = \frac{r}{h-r}\).

Key Algebraic Identities

\[(h - r)(\tan\gamma_{n+1} - \tan\gamma_n) = r(\sec\gamma_n + \sec\gamma_{n+1})\]

\[\tan\gamma_n = \sinh(a + n b), \qquad \sec\gamma_n = \cosh(a + n b), \qquad \frac{r}{h - r} = \tanh\frac{b}{2}\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.19 Equal Incircles Theorem — Folios (Folios 0125–0126, 2 scanned manuscript pages in resources/seelig/19_equal-incircle-theorem)
Euclid’s Elements	IV.4, VI.4 (EuclidElements)
GEOMETOR Elements (G-Index)	G.3 extremities of a segment
Encyclopedia of Triangle Centers	X(1) — Incenter (Kimberling1998)
Seelig Spreadsheets	Incircle-Incenter (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.4 Carnot’s Theorem, F.28 Incenter