Fig. 180.   Circle Divided into Eight Equal Parts by the Use of a 22 1/2x67 1/2 degree Triangle.

Fig. 180. - Circle Divided into Eight Equal Parts by the Use of a 22 1/2x67 1/2-degree Triangle.

A similar use of the 30 X 60-degree triangle is shown in Fig. 182, by which a circle is divided into six equal parts. Bring the blade of the T-square tangent to or near the circle, as shown by A B. Then place the set-square as shown by G B M, bringing the side G B against the center of the circle, drawing the line D L. Then place it as shown by the dotted lines, bringing the side A H against the center, scribing the line F E. Then, by reversing the set-square, placing the side G M against the straight-edge, erect the perpendicular C I, completing the division. The following are a few of the problems to which these principles may be advantageously applied.

48. To Inscribe an Equilateral Triangle within a Given Circle. - In Fig. 183, let D be the center of the given circle. Set the side E F of a 30-degree set-square against the T-square, as shown, and move it along until the side E G touches D. Mark the point B upon the circumference of the circle. Reverse the set-square so that the point E will come to the right of the side F G and move it along in the reversed position until the side E G again meets the point D, and mark the point C. Now move the T-square upward until it touches the point D, and mark the point A. Then A B and C are points which divide the circle into three equal parts. The triangle may be easily completed from this stage by drawing lines connecting A B, B C and C A, with any straight-edge or rule, but greater accuracy is obtained by the further use of the set-square, as follows: Place the side F G of the set-square against the T-square, as shown in Fig. 184, and move it along until the side E G touches the points A and C, as shown. Draw A C, which will be one side of the required triangle. Set the side E F of the set-square against the T-square, and move it along until the side F G coincides with the points C and B. Then draw C B, which will be the second side of the triangle.

Fig. 181.   Proper Method of Using a 45 degree Triangle.

Fig. 181. - Proper Method of Using a 45-degree Triangle.

Fig. 18S   Method of Using a 30 x 60 degree Triangle in Dividing the Circle.

Fig. 18S - Method of Using a 30 x 60-degree Triangle in Dividing the Circle.

Place the side F G of the set-square against the T-square, with the side E F to the right, and move it along until the side E G coincides with the points A and B. Then draw A B, thus completing the figure. The same results may be accomplished with less work-by first establishing the point A by bringing the T-square against the center, and then using the set-square, as shown in Fig. 184. The different methods are here given in order to more clearly illustrate the use of the tools employed.

The Construction Of Regular Polygons II By The Use 186

Fig. 183.

The Construction Of Regular Polygons II By The Use 187

Fig. 184.

To Inscribe an Equilateral Triangle within a Given Circle.

49. To Inscribe a Square within a Given Circle Let D, in Fig. 185, be the center of the given circle. Place the side E F of a 45-degree set-square against the T-square, as shown, and move it along until the side E G meets the point D. Mark the points A and B. Reverse the set-square, and in a similar manner mark the points C and H. The points A, H, B and C are corners of the required square. Move the T-square upward until it coincides with the points A and H and draw A H, as shown in Fig. 186. In like manner draw C B. With the side E F of the set-square against the T-square, move it along until the side G F coincides with the points B and H, and draw B H. In a similar manner draw C A, thus completing the figure. 50. To Inscribe a Hexagon within a Given Circle. - In Fig. 187, let O be the center of the given circle. Place the side E F of a 30-degree set-square against the T-square, as shown. Move the set-square along until the side E G meets the point O. Mark the points A and B. Reverse the set-square, and in like manner mark the points C and D. With the side F G of the set-square against the 1 -square, move it along until the side E F meets the point O, and mark I and H. Then A, H, D, B, I and C represent the angles of the proposed hexagon. From this stage the figure may be readily finished by drawing the sides by means of these points, using a simple straight-edge; but greater accuracy is attained in completing the figure by the further use of the set-square, as shown in Fig. 188. With the side E F of the set-square against the T-square, as shown, draw the line H D, and by moving the T-square upward draw the side CI. Reversing the set-square so that the point F is to the left of the point E, draw the side A H, and also, by shifting the T-square, the side I B. With the edge E F of the set-square against the T-square, move it up until the side G F coincides with the points B and D, and draw the side B D. In like manner draw A C, thus completing the figure. In this figure, as with the triangle, the same results may be reached by establishing the points H and I, by means of a diameter drawn at right angles to the T-square, as shown in the engravings, and, using it as a base, employing the set-square, as shown in Fig. 188. The first method shown is, however, to be preferred in many instances, on account of its great accuracy.