F.48 Verisimilitude, Corner Subtriangles, and Excircle Homothety

Incircle Tangent Subtriangles (r_A + r_B + r_C = r), Excircle Similitude, and Cevian Bounds

If lines parallel to the sides of triangle ABC are drawn tangent to the incircle, the three corner subtriangles are homothetic to ABC with scale factors x/s, y/s, z/s (where x = s - a, y = s - b, z = s - c), so their inradii satisfy r_A + r_B + r_C = r.

Interactive GEOMETOR construction for F.48 (Verisimilitude, Corner Subtriangles, and Excircle Homothety). 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_A = foot(I, line(B, C)) c_in = circle(I, T_A) P_A = intersection(segment(A, I), c_in) l_tan = perpendicular(P_A, line(A, I))

Mathematical Synthesis

In this nine-folio investigation (SeeligNotebooks), Seelig explores homothetic scaling (verisimilitude) across three connected settings. First (Folios 1–4), drawing tangents to the incircle parallel to \(BC, CA, AB\) cuts off three corner subtriangles homothetic to \(\triangle ABC\) from vertices A, B, C. Because the distance from A to the incircle touchpoint is \(x = s-a\), the corner subtriangle at A has semiperimeter \(s_A = x = s-a\)! Consequently its scale factor is \(x/s\), its inradius is \(r_A = r(x/s)\), its area is \(K_A = K(x/s)^2\), and summing over all three corners gives the remarkable invariant \(r_A + r_B + r_C = r\frac{x+y+z}{s} = r\).

Second (FWS 2014/9/7, Folios 5–6), Seelig analyzes :term:`CENTERS OF SIMILITUDE`: :term:`EXCIRCLES <excircle>`, treating vertex A as the external center of similitude of the incircle \((I, r)\) and the \(A\)-excircle \((I_A, R_A)\) separated by \(d = |II_A|\), deriving \(|AI| = d\frac{r}{R_A - r}\) and \(|AI_A| = d\frac{R_A}{R_A - r}\). Third (Folios 7–9), he solves Stewart's theorem as a quadratic in \(k\) for a prescribed cevian length \(d = 4\) in \((a,b,c) = (7,5,10)\), finding the two roots \(k \in \{0.613, 0.147\}\) above the altitude minimum \(d_{\min} = h_c = 3.25\).

Key Algebraic Identities

\[s_A = x = s - a, \qquad r_A = r\frac{x}{s}, \qquad K_A = K\left(\frac{x}{s}\right)^2, \qquad r_A + r_B + r_C = r\frac{x + y + z}{s} = r\]

\[|AI| = |II_A|\frac{r}{r_a - r}, \qquad |AI_A| = |II_A|\frac{r_a}{r_a - r}, \qquad c^2 k^2 - (c^2 + a^2 - b^2) k + (a^2 - d^2) = 0\]

Cross-Repository Canon & Archival Provenance

Seelig Archive Folios	F.48 Verisimilitude, Corner Subtriangles, and Excircle Homothety — Folios (Folios 0306–0314, 9 scanned manuscript pages in resources/seelig/48_verisimilitude)
Euclid’s Elements	IV.4, VI.4 (EuclidElements)
GEOMETOR Elements (G-Index)	G.5 surface, G.8 boundary
Encyclopedia of Triangle Centers	X(1) — Incenter, X(55) — Internal Center of Similitude of Incircle and Circumcircle, X(56) — External Center of Similitude of Incircle and Circumcircle (Kimberling1998)
Seelig Spreadsheets	Centers of Similitude, Incircle-Incenter, Stewart's Theorem (Spreadsheets)

Constructive Dependency Graph

Prerequisites: F.5 Centers of Similitude, F.14 Conway Notation, F.29 Inradius, F.46 Stewart’s Theorem