How to build a regular square inscribed in a circle

You can easily enter a square into a circle using drawing tools. But this problem is solved even with their complete absence. It is only necessary to remember some properties of the square.

You will need

-compass
-pencil
-gon
-scissors

Instruction manual

Draw a sketch for the task. Obviously, the diameter of the circle is the diagonal of the square inscribed in this circle. Remember the well-known property of a square: its diagonals are mutually perpendicular. Use this diagonal relationship when constructing a given square.

Draw a diameter in a circle. From the center, use a square to draw the second diameter at an angle of 90 degrees to the first. Connect the intersection points of the perpendicular diameters with the circle and get a square inscribed in this circle.

If you only have compasses from drawing instruments, draw a circle. Mark an arbitrary point on the circle and draw a diameter through it using any object with a smooth edge. Now you need to use a compass to divide half the circle between the ends of the diameter into two equal parts. From the points of intersection of the diameter with the circle, make two notches, keeping the compass solution unchanged. Draw a second diameter through the intersection point of these serifs and the center of the circle. Obviously, it will be perpendicular to the first.

If you don’t have any drawing tools, you can use a pair of scissors to cut out a circle from paper, limited by a given circle. Fold the cut shape exactly in half. Repeat the operation. It is necessary to combine the ends of the fold line, then the curved sections coincide without additional effort. Lock the addition lines. Now expand the circle. The fold lines are clearly visible. Fold the segments of the circle between the intersection points of the fold lines with the circle and cut off these segments. Cut lines are sides of the desired square. Place the cut square in the given circle, aligning its center with the intersection point of the fold lines of the circle. The vertices of the square will be lying on the circle, which was required to be performed.

You can easily enter a square into a circle using drawing tools. But this problem is solved even with their complete absence. It is only necessary to remember some properties of the square.

You will need

-compass
-pencil
-gon
-scissors

Instruction manual

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So, to build a square inscribed in a circle, we build a “Vesica Piscis” (Fig. 1).

1. We build perpendiculars “Vesica Piscis” AB and CD (Fig. 1). A and B are the centers of the original circles, C and D are the points of their intersection. We mark the point of intersection of the perpendiculars AB and CD, point O.

2. Build a circle inscribed in “Vesica Piscis”, centered at point O and with a radius equal to half the radius of any of the original circles, for example AO.

3. Mark the points of intersection of the perpendicular CD and the inscribed circle (with center O, radius AO), points F and E.

4. Draw direct AF and AE (Fig. 2).

5. Mark the points of intersection of the straight lines AF, AE with the original circle (with center A, radius AB), points G, H, K, L.

6. Draw the lines GL, LK, KH, HG (Fig. 3).

7. The square GHKL inscribed in the original circle (with center A, radius AB) is built.

Probably not at all superfluous, it will be recalled the basic properties of the square.

So.

1. The sides, the square, are equal.

2. Inside corners, squares, straight lines (90 0).

3. Diagonals, squares, are equal.

4. Diagonals, squares, are the bisectors of its angles.

5. Diagonals, squares, are perpendicular to each other.

6. Diagonals, squares, divide each other in half.

7. The centers of the inscribed and described squares coincide with the center of the corresponding circle.

8. The diagonals of the inscribed or described squares divide the circle into four equal parts.

Well, perhaps that's all ...