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CompassesA compass is a tool used to make a circle, and in making circles it can be utilized in many ways. Compasses are also used to transfer measurements, find a center points, and create right angles. Sometimes variations on the compass theme can be utilized as well to create accurate ellipses, arcs and spirals. Often underestimated, a compass is often an extremely useful tool. 
Geometrically, a circle is defined as the set of all points equal in distance from a single central point. All compasses create circles by first describing a radius, that equal distance between the circle and its center. The radius can be used to designate like measurements when necessary. The following examples are meant to show how to do a few simple constructions with a compass. No proofs are offered here.
Bisect a Line Segment
To bisect means to split in two equal parts. In this case, using a compass will allow one to not only define a center point, but to create true right angles. Variations on this theme can be used quite efficiently for many geometric tasks.

First, make a straight line with which to work. It can either be a random line, or one that is carefully placed. 

Place a couple of dots on the line to define endpoints of a segment. These dots can be carefully measured if necessary, or arbitrary if it does not really matter. 

Create two circles using the endpoints as foci. The entire circle is not necessary, as will be made evident next. To bisect the line, both circles must have the same radius. If you only wish to create a right angle, and not find a midpoint, then the circles can be of different sizes. 

Use a straight edge to draw a second line through the two points where the circles intersect. This line will cross the original at a perfect right (90 degree) angle. If the circles have the same radius, the second straight line will also pass through the exact mid point on the line segment. 

This method can be used to create right angles through specific points as well. just place a point on a line through which you wish a right angle intersection. Measure an equal distance to either side of that point and use the measured points as endpoints. 
Make a Square (example)
First make a 90 degree angle with pencil lines. 

Placing the focus of the compass at the intersection, mark equal distances on both lines. 

Complete the square at those points making right angles with a compass. 

Remove any extra lines with an eraser. Each side has an equal measure. 
Bisect an Angle
Line segments are not the only thing that a compass can spilt into equal halves. Angles can be bisected as well.
Begin with an angle. It can be arbitrary, or carefully measured. 

Use a compass to measure off equal distances along the "legs" of the angle. This is most easily done by drawing the arc of a circle with its focus at the angle intersection. 

Next, draw two circles of equal radius centering their foci at the two points where the first circle intersected the legs of the angle. 

Draw a line through the points where these two circles intersect. This line splits the angle exactly in two. 

Since any angle can be split in two this way, a compass can be used to create 90 degree angles from 180 degree straight lines, and 45 degree angles from 90 degree intersections. 

Find the Center of a Circle
It is not always a circle of which one needs to locate the focus. More often it is an arc, a curve, or partial circle. Still, the same basic rules apply.
Start by describing a circle. If only a section of it appears in a picture, it helps to imagine the whole thing present. 

A polygon is inscribed in a circle when each endpoint lies exactly on the edge of the circle. Any line which bisects one of the sides of such a polygon at a right angle will pass through the exact center of the circle. 

Knowing this, we only need two sides of a polygon to locate the center. Indeed, we only need two line segments with endpoints on the circle. 