To the question: how to divide a circle into three equal parts using a compass)? tell me this please!! given by the author Embassy the best answer is
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Let a circle of radius R be given. We need to divide it into three equal parts using a compass. Open the compass to the size of the radius of the circle. You can use a ruler, or you can place the needle of the compass in the center of the circle, and move the leg to the link describing the circle. In any case, the ruler will come in handy later.
Place the compass needle in a random location on the circumference of the circle, and with a stylus, draw a small arc intersecting the outer contour of the circle. Then install the compass needle at the found reference point and draw an arc again with the same radius (equal to the radius of the circle).
Repeat these steps until the next intersection point coincides with the very first one. You will get six links on circles spaced at equal intervals. All that remains is to select three points through one and use a ruler to connect them to the center of the circle, and you will get a circle divided into three.
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A circle can be divided into three parts if, using a compass, from the point of intersection of a straight line drawn through the center of the circle O, make with a compass notches B and C on the line of the circle with a value equal to the radius of this circle.
Thus, two required points will be found, and the third is the opposite point A, where the circle and straight line intersect.
Further, if necessary, using a ruler and pencil

you can draw an embedded triangle.

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To mark into three parts we use the radius of the circle.

Turn the compass over backwards. Place the needle on
the intersection of the center line with the circle, and the stylus in the center. outline
an arc intersecting a circle.

The intersection points will be the vertices of the triangle.

Dividing a circle into six equal parts and constructing a regular inscribed hexagon is done using a square with angles of 30, 60 and 90º and/or a compass. When dividing a circle into six equal parts with a compass, arcs are drawn from two ends of the same diameter with a radius equal to the radius of the given circle until they intersect with the circle at points 2, 6 and 3, 5 (Fig. 2.24). By sequentially connecting the resulting points, a regular inscribed hexagon is obtained.

Figure 2.24

When dividing a circle with a compass, from the four ends of two mutually perpendicular diameters of the circle, an arc with a radius equal to the radius of the given circle is drawn until it intersects with the circle (Fig. 2.25). By connecting the resulting points, a dodecagon is obtained.

Figure 2.25

2.2.5 Dividing a circle into five and ten equal parts
and construction of regular inscribed pentagon and decagon

The division of a circle into five and ten equal parts and the construction of a regular inscribed pentagon and decagon is shown in Fig. 2.26.

Figure 2.26

Half of any diameter (radius) is divided in half (Fig. 2.26 a), point A is obtained. From point A, as from the center, draw an arc with a radius equal to the distance from point A to point 1 to the intersection with the second half of this diameter, at point B( Fig. 2.26 b ). Segment 1 is equal to a chord subtending an arc whose length is equal to 1/5 of the circumference. Making notches on the circle (Fig. 2.26, in ) radius TO equal to segment 1B, divide the circle into five equal parts. Starting point 1 is chosen depending on the location of the pentagon. From point 1, build points 2 and 5 (Fig. 2.26, c), then from point 2, build point 3, and from point 5, build point 4. The distance from point 3 to point 4 is checked with a compass. If the distance between points 3 and 4 is equal to segment 1B, then the construction was carried out accurately. It is impossible to make serifs sequentially, in one direction, as errors occur and the last side of the pentagon turns out to be skewed. By sequentially connecting the found points, a pentagon is obtained (Fig. 2.26, d).

Dividing a circle into ten equal parts is carried out similarly to dividing a circle into five equal parts (Fig. 2.26), but first divide the circle into five parts, starting construction from point 1, and then from point 6, located at the opposite end of the diameter (Fig. 2.27, A). By connecting all the points in series, they get a regular inscribed decagon (Fig. 2.27, b).

Figure 2.27

2.2.6 Dividing a circle into seven and fourteen equal parts
parts and construction of a regular inscribed heptagon and
quadragon

The division of a circle into seven and fourteen equal parts and the construction of a regular inscribed heptagon and a fourteen-sided triangle are shown in Fig. 2.28 and 2.29.

From any point on the circle, for example point A , draw an arc with the radius of a given circle (Fig. 2.28, a ) until it intersects with the circle at points B and D . Let's connect the points Vi D with a straight line. Half of the resulting segment (in this case segment BC) will be equal to the chord that subtends an arc constituting 1/7 of the circumference. With a radius equal to the segment BC, notches are made on the circle in the sequence shown in Fig. 2.28, b . By connecting all the points in series, they get a regular inscribed heptagon (Fig. 2.28, c).

Dividing the circle into fourteen equal parts is done by dividing the circle into seven equal parts twice from two points (Fig. 2.29, a).

Figure 2.28

First, the circle is divided into seven equal parts from point 1, then the same construction is performed from point 8 . The constructed points are connected sequentially by straight lines and a regular inscribed quadrangle is obtained (Fig. 2.29, b).

Figure 2.29

Construction of an ellipse

The image of a circle in a rectangular isometric projection in all three projection planes is an ellipses of the same shape.

The direction of the minor axis of the ellipse coincides with the direction of the axonometric axis, perpendicular to the projection plane in which the depicted circle lies.

When constructing an ellipse depicting a circle of small diameter, it is enough to construct eight points belonging to the ellipse (Fig. 2.30). Four of them are the ends of the ellipse axes (A, B, C, D), and the other four (N 1, N 2, N 3, N 4) are located on straight lines parallel to the axonometric axes, at a distance equal to the radius of the depicted circle from the center ellipse.

1. BRIEF THEORETICAL INFORMATION

1.1. Geometric constructions

Dividing a circle into equal parts

Some parts have elements evenly distributed around the circumference. When making drawings of parts that have similar elements, you must be able to divide the circle into equal parts. Techniques for dividing a circle into equal parts are shown in Fig. 1

Rice. 1. Dividing a circle into equal parts

With sufficient accuracy, you can divide the circle into any number of equal parts using the table of coefficients for calculating the stroke length.

Based on the number of equal segments on the circle (Table 1), we find the corresponding coefficient. By multiplying the resulting coefficient by the diameter of the circle, we obtain the length of the chord, which we plot on the circle with a compass.

Table 1 - Coefficient for determining chord length

Number of parts of a circle	Coefficient

Making a mate between two lines

When drawing the contours of technical details and in other technical constructions, it is often necessary to perform conjugations (smooth transitions) from one line to another. The conjugation of two sides of an angle with an arc specified by the arc radius R is performed in the following sequence:

- two auxiliary straight lines are drawn parallel to the sides of the angle at a distance equal to R;

- the point of intersection of these lines will be the center of conjugation;

- from the center of the mate, perpendiculars are made to the given straight lines;

- the points of intersection of perpendiculars with given lines are called conjugation points;

- an arc of radius R is built from the center of the mate, connecting the mate points.

In Fig. 2 shows examples of constructing mates when the radius of the mate arc is specified. In this case, it is necessary to determine the fillet center and fillet points. The contour of the part is traced using a compass.

Rice. 2. Techniques for constructing connections

In technology, it is often necessary to draw curved lines made up of a large number of small arcs of circles with a gradual change in the radius of their curvature. Such lines cannot be drawn with a compass. These curves are drawn using patterns and are called patterns. It is necessary to study the pattern of formation of a pattern curve and plot a number of points belonging to it on the drawing. The points are connected by a smooth curve with a thin line by hand, and the outline is done using a pattern.

To trace pattern curves you need to have a set of several patterns. Having chosen a suitable pattern, adjust the edge of part of the pattern to as many points as possible. To circle

the next section, you need to adjust the edge of the pattern to two or three more points, while the pattern should touch part of the already outlined curve. The method of drawing a curve along a pattern is shown in Fig. 3.

Rice. 3. Construction of a curve according to the pattern.

In Fig. Figure 4 shows an example of constructing an ellipse along given axes

Rice. 4. Construction of an ellipse

In Fig. Figure 5 shows an example of constructing a parabola by dividing the sides of angle AOC into the same number of equal parts. In Fig. Figure 6 gives an example of constructing an involute of a circle. Given

the circle is divided into 12 equal parts. Tangents to the circle are drawn through the division points. On the tangent drawn through point 12, the length of this circle is plotted and divided into 12 equal parts. Starting from point l on the tangents to the circle, segments equal to 1/12 of the circumference, 1/6, 1/4, etc. are successively plotted.

Rice. 5. Construction of a parabola

Rice. 6. Construction of an involute

Rice. 7.Construction of a sinusoid

Fig.8 Construction of the Archimedes spiral

In Fig. Figure 7 shows the method for constructing a sinusoid. A given circle is divided into 12 equal parts; a straight line segment equal to the length of the unfolded line is divided into the same number of equal parts.

During renovations, you often have to deal with circles, especially if you want to create interesting and original decorative elements. You also often have to divide them into equal parts. There are several methods to do this. For example, you can draw a regular polygon or use tools known to everyone since school. So, in order to divide a circle into equal parts, you will need the circle itself with a clearly defined center, a pencil, a protractor, as well as a ruler and compass.

Dividing a circle using a protractor

Dividing a circle into equal parts using the above-mentioned tool is perhaps the simplest. It is known that a circle is 360 degrees. By dividing this value into the required number of parts, you can find out how much each part will take (see photo).

Next, starting from any point, you can make notes corresponding to the calculations performed. This method is good when you need to divide a circle by 5, 7, 9, etc. parts. For example, if the shape needs to be divided into 9 parts, the marks will be at 0, 40, 80, 120, 160, 200, 240, 280 and 320 degrees.

Division into 3 and 6 parts

To correctly divide a circle into 6 parts, you can use the property of a regular hexagon, i.e. its longest diagonal must be twice the length of its side. To begin with, the compass must be stretched to a length equal to the radius of the figure. Next, leaving one of the legs of the tool at any point on the circle, the second one needs to make a notch, after which, repeating the manipulations, you will be able to make six points, connecting which you can get a hexagon (see photo).

By connecting the vertices of the figure through one, you can get a regular triangle, and accordingly the figure can be divided into 3 equal parts, and by connecting all the vertices and drawing diagonals through them, you can divide the figure into 6 parts.

Division into 4 and 8 parts

If the circle needs to be divided into 4 equal parts, first of all, you need to draw the diameter of the figure. This will allow you to get two of the required four points at once. Next, you need to take a compass, stretch its legs along the diameter, then leave one of them at one end of the diameter, and make the other notches outside the circle from below and above (see photo).

The same must be done for the other end of the diameter. After this, the points obtained outside the circle are connected using a ruler and pencil. The resulting line will be a second diameter, which will run clearly perpendicular to the first, as a result of which the figure will be divided into 4 parts. In order to get, for example, 8 equal parts, the resulting right angles can be divided in half and diagonals drawn through them.

When performing graphic work, you have to solve many construction problems. The most common tasks in this case are dividing line segments, angles and circles into equal parts, constructing various conjugations.

Dividing a circle into equal parts using a compass

Using the radius, it is easy to divide the circle into 3, 5, 6, 7, 8, 12 equal sections.

Dividing a circle into four equal parts.

Dot-dash center lines drawn perpendicular to one another divide the circle into four equal parts. Consistently connecting their ends, we get a regular quadrilateral(Fig. 1) .

Fig.1 Dividing a circle into 4 equal parts.

Dividing a circle into eight equal parts.

To divide a circle into eight equal parts, arcs equal to a quarter of the circle are divided in half. To do this, from two points limiting a quarter of the arc, as from the centers of the radii of a circle, notches are made beyond its boundaries. The resulting points are connected to the center of the circles and at their intersection with the line of the circle, points are obtained that divide the quarter sections in half, i.e., eight equal sections of the circle are obtained (Fig. 2 ).

Fig.2. Dividing a circle into 8 equal parts.

Dividing a circle into sixteen equal parts.

Using a compass, dividing an arc equal to 1/8 into two equal parts, apply notches to the circle. By connecting all the serifs with straight segments, we get a regular hexagon.

Fig.3. Dividing a circle into 16 equal parts.

Dividing a circle into three equal parts.

To divide a circle of radius R into 3 equal parts, from the point of intersection of the center line with the circle (for example, from point A), an additional arc of radius R is described as from the center. Points 2 and 3 are obtained. Points 1, 2, 3 divide the circle into three equal parts.

Rice. 4. Dividing a circle into 3 equal parts.

Dividing a circle into six equal parts. The side of a regular hexagon inscribed in a circle is equal to the radius of the circle (Fig. 5.).

To divide a circle into six equal parts, you need points 1 And 4 intersection of the center line with the circle, make two notches with a radius on the circle R, equal to the radius of the circle. By connecting the resulting points with straight line segments, we obtain a regular hexagon.

Rice. 5. Dividing a circle into 6 equal parts

Dividing a circle into twelve equal parts.

To divide a circle into twelve equal parts, the circle must be divided into four parts with mutually perpendicular diameters. Taking the points of intersection of the diameters with the circle A , IN, WITH, D beyond the centers, four arcs of the same radius are drawn until they intersect with the circle. Received points 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 and dots A , IN, WITH, D divide the circle into twelve equal parts (Fig. 6).

Rice. 6. Dividing a circle into 12 equal parts

Dividing a circle into five equal parts

From point A draw an arc with the same radius as the radius of the circle until it intersects with the circle - we get a point IN. Dropping the perpendicular from this point, we get the point WITH.From point WITH- the middle of the radius of a circle, as from the center, an arc of radius CD make a notch on the diameter, we get a point E. Line segment DE equal to the length of the side of the inscribed regular pentagon. Making it a radius DE serifs on the circle, we get the points of dividing the circle into five equal parts.

Rice. 7. Dividing a circle into 5 equal parts

Dividing a circle into ten equal parts

By dividing a circle into five equal parts, you can easily divide the circle into 10 equal parts. Drawing straight lines from the resulting points through the center of the circle to the opposite sides of the circle, we get 5 more points.

Rice. 8. Dividing a circle into 10 equal parts

Dividing a circle into seven equal parts

To divide a circle of radius R into 7 equal parts, from the point of intersection of the center line with the circle (for example, from the point A) are described as an additional arc from the center the same radius R- get a point IN. Dropping a perpendicular from a point IN- we get a point WITH.Line segment Sun equal to the length of the side of the inscribed regular heptagon.

Rice. 9. Dividing a circle into 7 equal parts