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Geometry and a Square - A game with right angles. Air Strike - Find a maximum set of points covered by two disks. Build the Fence - Convex hull, huh? Cell Phone - Cover maximum points with one disk. Transmitters - Sort points along the rotation angle. The Ant - Find a route for a little ant. Equipment Box - Does one rectangle fit into another? Lifting the Stone - A search for the center of mass.
Almost Shortest Path - Find an almost shortest path. Lego - Did you use to play Lego? Then this shouldn't be a challenge for you! Ant - Ants know nothing about matrix multiplication and fast exponentiation. Paradox - Identify a contradiction in a set of arguments. The Bottom of a Graph - Find a bottom of a graph.
http://www.spoj.com/problems/GEOPROB/ http://www.spoj.com/problems/JASIEK/ http://www.spoj.com/problems/CVXPOLY/ http://www.spoj.com/problems/VCIRCLES/ http://www.spoj.com/problems/AE5B1/ http://www.spoj.com/problems/RUNAWAY/ http://www.spoj.com/problems/CONDUIT/ http://www.spoj.com/problems/TETRA/ http://www.spoj.com/problems/QUADAREA/ http://www.spoj.com/problems/TRICENTR/ http://uva.onlinejudge.org/external/117/11796.html
You may also want to explore more basic and advanced geometric concepts using free online tool: http://www.geogebra.org/webstart/geogebra.html