The last thing I will include is a theorem from the Geometry book the we use in my classroom that I think would translate nicely into a problem for Geometer's Sketchpad. There is a theorem that is titled Perpendicular Bisector Theorem and it states: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoint of the segment.
I would have the kids do this first doing a compass and straightedge construction using Sketchpad to construct the perpendicular bisector. Later they could just use the Sketchpad command to create it. Then they would have to select several points on the bisector and measure the distance from each endpoint of the segement to the selected point. I would do this as a "discovery" activity meaning that kids would have to state the theorem after having done the investigation.
As an extension, I would ask them to do the Chain Up a Goat problem using Sketchpad. This is also known as the Locate the Radio Tower problem and has many other names. Kids are given the location of three trees and they have to find the place where the goat can be chained up in order for it to be able to reach each tree. Basically kids will find the center of the circle through these three points using the intersection point of the perpendicular bisectors of the three segments connecting these points. the circle given three points.