My geometry teacher challenged me to find a way to find the center of a given circle using only one circle and four lines. No tangents either. I can't seem to find a way to do this, all I've found so far are different ways of finding the diameter of the circle with one line left. I have always get one line left to draw, and through my own constructions,there aren't any points that I could use to draw another straight line through the diameter to find the center. This is Euclidean Geometry by the way.
A simpler version of the paragraph above is to simplify the construction of Book Three Proposition one in Euclidean Geometry so that the construction takes 4 straight lines and one circle.
Best Answer
The solution below is based on the assumption that drawing a line parallel to the other requires no auxiliary circle(s).
Step 0: Select a random point A on the given circle.
Step 1: Draw chord AB of reasonable length.
step 2: Draw $C_1$ (centered at A, radius = AB).
Step 3: Locate C where BA produced cuts $C_1$.
step 4: The two circles will meet at D. CD, when joined, will cut the given circle at E.
Step 5: Note that $\angle CDB = \angle EDB = 90^0$ because BC is the diameter of $C_1$.
Step 6: Join BE which is then the diameter of the given circle.
Step 7: Through A, draw a line parallel to CD cutting BE at F. By intercept theorem, FB = FE.
[Note:- Drawing a line parallel to a given line can be done by translating "set squares, or straight rulers" without drawing auxiliary circles. Whether the construction of such line is considered as a 1-step construction or not depends heavily on the rule of the game.]