of a line and a circle (if they intersect) Creating the one or two points in the intersection of two circles (if they intersect). #1 - A line through two points It is understood that a straight line cannot be drawn without a straightedge. A line is considered to be given by any two points, as any such pair define a unique line. In keeping with the intent of the theorem which we aim to prove, the actual line need not be drawn but for aesthetic reasons. #2 - A circle through one point with defined center This can be done with a compass alone. A straightedge is not required for this. #5 - Intersection of two circles This construction can also be done directly with a compass. #3, #4 - The other constructions Thus, to prove the theorem, only compass-only constructions for #3 and #4 need to be given. Notation and remarks The following notation will be used throughout this article. A circle whose center is located at point and that passes through point will be denoted by . A circle with center and radius specified by a number, , or a line segment will be denoted by or , respectively. In general constructions there are often several variations that will produce the same result. The choices made in such a variant can be made without loss of generality. However, when a construction is being used to prove that something can be done, it is not necessary to describe all these various choices and, for the sake of clarity of exposition, only one variant will be given below. However, many constructions come in different forms depending on whether or not they use circle inversion and these alternatives will be given if possible. It is also important to note that some of the constructions proving the Mohr-Mascheroni theorem require the arbitrary placement of points in space, such as finding the center of a circle when not already provided (see construction below). In some construction paradigms - such as in the geometric definition of the constructible number - this may be prohibited. In such a paradigm, however, for example, circles without their centers will not be provided by hypothesis, thus there is no issue. Some preliminary constructions To prove the above constructions #3 and #4, which are included below, a few necessary intermediary constructions are also explained below since they are