Alternative compass and straightedge construction My problem is motivated by software like Geogebra.
Suppose that we are allowed to use the following two tools only:


*

*given any two points, we can construct a line passing through both of them

*given any three non-collinear points, we can construct a circle passing through all of them.
I tried to use these two tools to do some simple geometric constructions, such as bisecting an angle or a line segment but failed to achieve anything. I wonder whether it is possible to do these.
Note: As mentioned in John Hughes's answer, I have to clarify my problem.
The setting of the problem is just like that in the classical compass and straightedge problem, but with the compass replaced by a 'three-point-one-cirlce' instrument.
 A: At the very least, you'll need a few other "hidden" operations: 


*

*Given any line, can you pick two distinct points on it? Three distinct points? 

*Can you pick three noncollinear points in the plane? 

*Given a circle, can you pick points on it?
Without these, the empty geometry and the geometry consisting of one line, which contains either 1 or 2 points, are both perfectly valid, and so is the geometry consisting of just the points $(0,0), (\pm 1, 0)$, and the three lines that join them (and the circle through them, which consists of all three points!). 
That latter geometry is interesting because the vertical and horizontal lines have no "angle bisector", which shows that if your axioms only allow the construction of lines and circle, constructing bisectors is impossible. (Also note that angle-measure is pretty dicey, too). 
If you mean "Starting from the Euclidean plane, with its notions of angle measure and distance measure, and the collection of lines guaranteed by the axioms of Euclidean geometry, can I, using just these two construction rules, produce lots of interesting things like bisectors, etc.?" then I suspect that you can do some stuff, but not a whole lot. 
But you really need to clarify your question a bit. 
