Geometric Brain Teasers

[MrFreeze]

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! :shock: 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 ?. Two points M and N are
chosen on ? 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 ? and m and a point P, use only ruler to construct
the line through P parallel to ? 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.)

 

[Deleted member 4993]

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! :shock: 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 ?. Two points M and N are
chosen on ? 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 ? and m and a point P, use only ruler to construct
the line through P parallel to ? 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.)

Please share your work with us, indicating exactly where you are stuck - so that we may know where to begin to help you.

 

[MrFreeze]

its not so much being stuck at a certain point; its that i have tried different approaches, but then it gets me nowhere. also, i was told that number 4 requires the use of 30 lines...

 

[mmm4444bot]

its not so much being stuck

i have tried different approaches

[they get] me nowhere

To me, it seems like you're stuck.

Perhaps your approaches would get us somewhere. We won't know, until after you tell us what you've already done.

PS: We prefer a separate thread for each exercise; otherwise, a discussion of all seven exercises at once could easily lead to a awkward thread of dozens of posts spread over several pages.