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.