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This paper presents the open-source robot inverse kinematics (IK) solver IK-Geo, the fastest general IK solver based on published literature. In this unifying approach, IK for any 6-DOF all-revolute (6R) manipulator is decomposed into six canonical geometric subproblems solved by intersecting circles with other geometric objects. We present new efficient and singularity-robust solutions to these subproblems using geometric and linear algebra methods. IK-Geo finds all IK solutions including singular solutions and sometimes least-squares solutions by solving for subproblem solutions in all cases, including in a continuous and sometimes least-squares sense when a solution does not exist. Robots are classified into kinematic families based on cases of intersecting or parallel joint axes, and robots in the same family use the same IK algorithm. 6R robots with three intersecting or parallel axes are solved in closed form, and all solutions are found exactly without iteration. Other 6R robots are efficiently solved by searching for zeros of an error function of one or two joint angles. The subproblem and IK solutions are easy to understand, implement, test, and modify, meaning this method is readily ported to new languages and environments. We connect our geometric method with less efficient but more robust polynomial-based methods: rather than using search, subproblems and error functions may be written in terms of the tangent half-angle of one joint. This results in a system of multivariate polynomial equations from which the univariate polynomial with zeros corresponding to IK solutions is readily derived.