This paper presents a geometric method to solve the inverse kinematics
for a class of 3-joint placeable robotic manipulators.  A distinct
feature of the method is that a mechanical geometric reasoning process
is used for replacing conventional algebraic calculations. The
conventional algebraic method solves the problem by finding analytic
solutions of arm equations of robotic manipulators.  Arm equations are
usually polynomial equations, algebraic calculations in finding the
solutions are always very complicated, and even fail for some cases of
3-joint non-loose placeable manipulators [P91].  The geometric method
presented in this paper specifies manipulators in terms of vectors,
instead of arm equations, and employ reasonings about geometric
variables such as volume, area, length and Pythagoras difference,
instead of algebraic calculations. It can decide whether a 3-joint
placeable manipulator is loose, and can mechanically find out analytic
solutions of the inverse kinematics for any non-loose one.  The method
is therefore simple and understandable.