Math Is Fun Forum

I have a series of questions that I have been trying to solve and I need help with the solutions, they are driving me crazy! Each question needs to be proven also, not just constructed. Here they are:

1. Describe all isometries of the Manhattan plane.

2. Two points A and B lie on one side of line ℓ. Two points M and N are
chosen on ℓ such that AM +BM is minimal and AN = BN. Show that points
A, B, M and N lie on one circle.

3. Given two parallel lines ℓ and m and a point P, use only ruler to construct
the line through P parallel to ℓ and m.

4. Given two concentric circles construct their center using only ruler.

5. Show that any construction with only ruler can be done with a “short ruler”;
i.e. an instrument which makes possible to draw a line only through sufficiently
close pair of points.

6. Assume you have an instrument which makes possible to draw a circle or
line through any given three points. Show that it is impossible to construct the
center of given circle using only this instrument.

7. Give a ruler-and-compass construction of a circle or a line which perpendicular
to each of three given circles. (You may assume any two of three given
circles do not intersect.)