How to split a circle?

Divide the circle into an equal number of parts -this is not an empty occupation, necessary only to complicate your life and to hammer an unhappy head with a heap of information. From a practical point of view, this is useful for constructing regular polygons, pie charts, drawing stars. And you can just share the birthday cake (joke).

It would seem that the question of how to dividea circle on a part, itself the answer arises: "With the help of a protractor". But there are ways to do this very accurately, without resorting to mathematical calculations, but simply by using a compass, ruler and pencil. First of all, we agree that under our constructions we will use such a notion as center lines. These are lines intersecting in the center of a circle of radius R (point O) at an angle of 90 degrees. And the points where these lines intersect with the circle are numbered clockwise 1, 2, 3 and 4. So, between the cases, we were able to answer the question of how to divide the circle into 4 parts.

Now find out how to divide the circle by 3equal parts. The first desired point is point 1. Now fix the radius of the divisible circle on the compass and put its needle at point 3, make two notches on the circle. So we find two more points that, together with point 1, will be the solution of the problem of dividing the circle into three parts. Now, using the same principle, you can divide the circle into 12 parts. To do this, make two notches of radius R on the circle in sequence, placing the circle needle in each of the points 1, 2, 3 and 4. The eight intersection points of the notches with a circle, together with points 1, 2, 3 and 4, divide it into 12 parts .

To solve the problem of how to divide a circle by 5parts, you first need to figure out how to split a piece in half with a compass. To do this, we fix the length of the divisible segment on the compass, and draw two circles with the centers at the ends of the segment. Then we connect two points in which these circles intersect a line. This straight line will divide our segment in half.

Now, armed with this knowledge, let's move on topreviously assigned problem of dividing the circle into 5 equal parts. We divide the segment between the center of the circle O and the point 4 in half. We obtain a point E. Now, by radius E1 we make a notch on the segment O2. The point of intersection of the intersection with the segment O2 is called F. The segment EF is the length of the side of the pentagon inscribed into the circle, and consequently its vertices will divide our circle into 5 parts. From any point on the circle we construct an arc of radius EF, defining one its intersection with the circle. The subsequent construction is carried out successively from each newly constructed point. With a high accuracy of construction, the last point coincides with the initial one. The resulting points will divide the circle into 5 equal parts.