Problem 3. On a plane, there are 3 different points M, G, and T such that the figure formed by points M, G, and T has no axis of symmetry. Construct a point U on this plane so that the figure formed by points M, G, T, and U has at least one axis of symmetry. How many different such points exist in this plane?
# Solution:

If points M, G, and T are the vertices of a right triangle, then there are 5 such points; otherwise, there are 6. Indeed, 3 such points are the points symmetric to the given ones with respect to the line containing the other two, since all three points do not lie on the same line (by the condition). For example, the point symmetric to M with respect to the line GT, and so on. Another 3 such points are the points symmetric to the given ones with respect to the line that is the perpendicular bisector of the segment with vertices in the other two. For example, the point symmetric to M with respect to the perpendicular bisector of the segment GT, and so on. For a right triangle, two of the three points coincide.