# Geometric Constructions 12

1. Given the midpoint C of a line segment AB, construct a line parallel to AB passing through a given point D using ruler only.
2. The straight lines l and m are parallel, construct the midpoint of A and B on m using ruler only.
3. Through the point A lying outside the given straight line l and m, draw a straight line parallel to the given ones using ruler only.
4. Given A, B on m and l parallel to m, construct C on m with AB = BC using ruler only.
5. Given a circle and its center, construct an inscribed square using ruler only.
6. Given two points, construct its midpoint using the compass alone.
7. Construct the inversion of a point with respect to a circle using the compass alone. (You may assume the point lies outside the circle.)
8. Construct the inversion of a straight line with respect to a circle using the compass alone. (You may assume the line lies outside the circle.)
9. Let circle A passes through the center of circle B. Construct the inversion of A with respect to B using the compass alone. (Find two distinct points of the inversion.)
10. Suppose that circle A does not pass through the center of circle B. Construct the inversion of A with respect to B using the compass alone.
11. Construct the circumcircle of a triangle using the compass alone.
12. Given a circle but not its center, find its center using the compass alone.
13. Divide a circle into four equal parts in five steps using the compass alone.
14. Given one side, to construct the other two corners of the square using the compass alone.
15. Given a diagonal, to construct the square using the compass alone.
16. Construct the vertices of a regular pentagon using the compass alone.

## Related Sites

http://www.math.niu.edu/~rusin/known-math/96/mascheroni
http://www.math.niu.edu/~rusin/known-math/97/mascher
http://forum.swarthmore.edu/dr.math/problems/bau11.15.95_2.html
WHO WAS GEORG MOHR?
Mascheroni's construction is taught in C&O 408 "History of Mathematics"
MATH 343 Projective and Inversive Geometries
64513 The Joys of Geometry
Biography of Lorenzo Mascheroni
Biography of Georg Mohr