Well, the first problem is defining "line" and "circle" in your new topology, and this in itself tends to complicated not just the answer, but the question. For

Well, the first problem is defining "line" and "circle" in your new topology, and this in itself tends to complicated not just the answer, but the question. For example, on the surface of a sphere, "lines" are a special case of circles, and all circles have two possible center points with two radii, which are distinct iff the circle is not a line. Once you've defined circles and lines, constructable sets looks very different. For example, you can always construct an absolute measure of distance, but the concept of parallel line segments doesn't exist, although you can have circles that are parallel to lines. So can you trisect an angle in this slightly exotic topology? There's a good chance that you can, actually. There's a construction in the plane for trisecting a segment. If that construction is still valid on the sphere, you can take your angle and re-construct the legs with length equal to 1/4 of the sphere's circumference. Now, the line segment connecting the endpoints of these legs is an arc of a circle whose center is your angle. Trisecting this segment, and connecting the trisection back to the angle, will trisect the angle.