{"id":"triangle","domain":"geometry","type":"Concept","title":"Triangle","slug":"triangle","url":"/mathematics/geometry/triangle/","summary":"The simplest closed polygon. Rigid shape (basis of structural engineering). Foundation of trigonometry. Rich classical geometry: Euler line, nine-point circle.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Coxeter, H.S.M. and Greitzer, S.L. (1967). Geometry Revisited. Mathematical Association of America."},{"tier":1,"citation":"Hartshorne, R. (2000). Geometry: Euclid and Beyond. Springer."},{"tier":2,"citation":"Johnson, R.A. (1929/2007). Advanced Euclidean Geometry. Dover."},{"tier":3,"citation":"Posamentier, A.S. and Salkind, C.T. (1996). Challenging Problems in Geometry. Dover."}],"relationships":[{"type":"proves","target":"pythagorean-theorem","label":"Pythagorean Theorem"},{"type":"uses","target":"brahmasphutasiddhanta","label":"Brāhmasphuṭasiddhānta — Heron's formula"},{"type":"built_on","target":"euclidean-geometry","label":"Euclidean Geometry"},{"type":"related_to","target":"group","label":"Group — triangle groups"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.8"}