CGAL
C++. Exact predicates + kernels. Delaunay, meshing, polytopes, Boolean ops. GPL/commercial dual license.
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GEOS
Open-source spatial algorithms. Powers PostGIS. GEOS = Geometry Engine, Open Source.
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Boost.Geometry
Header-only C++. Predicates, R-tree, spatial queries. MIT-like. Widely used.
Shapely (Python)
Python wrapper around GEOS. De facto standard for GIS in Python.