N-point K-person polygon triangulation — min/max diagonals, all angles (actual trigonometry)

K persons hold K knots of the rope → a K-gon. Slide K. Green = a MIN triangulation (K−3 diagonals, K−2 triangles). The faint lines are the full diagonal set (MAX = K(K−3)/2). Number of triangulations = Catalan(K−2). Motive: build triangulated G(V,E) BOLS to prove identities Euclidean-ly.

K persons

MIN R = K−3 (triangulation) · MAX R = K(K−3)/2 (all diagonals)
triangles = K−2 · #triangulations = Catalan(K−2)
all angles rational-cosine

A1134: integral = perimeter (A172) OR cumulative-toothpick (A288); derivative = perp-on-complement (A1133)/tip (A90)/angle (A89)

DBT-13 which triangulation does least-action pick? DBT-14 ear vs fan?