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THE POPULAR EDUCATOR. 



to the utmost extent that the limits of that figure will per- 

 mit ; and in the third the prefix circum gives the word the 

 meaning of drawing one figure round another. Inscription and 

 circumscription indicate operations that are precisely the reverse 

 of each other. 



PROBLEM XLII. In a given circle to inscribe a triangle equi- 

 angular to a given triangle. 



Let ABC (Fig. 61) be the given triangle, and DBF the given 

 circle : it is required to inscribe in the given circle DBF a triangle 



equiangular to the 

 given triangle ABC. At 

 any point D in the cir- 

 cumference of the circle 

 DBF draw x T as a 

 tangent to the circle, 

 and at the point D in 

 the straight line x Y 

 make the angle Y D F 

 equal to the angle ABC, 

 and the angle x D E 

 equal to the angle A c B ; and let the straight lines D E, D F 

 cut the circumference of the circle D E F in the points E and F. 

 Join E F. The triangle DBF, inscribed in the circle D E F, 

 is equiangular to the given triangle ABC. 



If it is desired to cut off a segment of a circle that shall con- 

 tain an angle equal to a given angle, as in the above figure 

 to cut off from the circle D E F a segment that shall con- 

 tain an angle equal to the angle ABC, all that we have to do 

 ia to draw a tangent to the circle, and at the point of con- 

 tact make an angle equal to the given angle, as the angle Y D F 

 was made equal to the angle ABC. The leg D F of the 

 angle Y D F must then be produced far enough to cut the 

 circumference of the circle D E F in the point F. Any angle 

 that may then be formed by drawing straight lines from D and 

 ! to any point in the segment, as the angle D E F or the angle 

 D F, is equal to the given angle ABC. 



PROBLEM XLIII. About a given circle to circumscribe a tri- 

 angle equiangular to a given triangle. 



Let ABC (Fig. 62) be the given triangle, and D E F the given 

 circle about which it is required to circumscribe a triangle equi- 

 angular to the given triangle 



ABC. 



Produce B c, the base of 

 the triangle ABC, both ways 

 to x and Y. Draw K L touch- 

 ing the circle D E F in the 

 point E, and from the centre 

 G of the circle DBF draw 

 the straight line G E per- 

 pendicular to K L. Then at 

 the point G, in the straight 

 line E G, make the angle 

 EOF equal to the angle 

 A c Y, and the angle E G D equal to the angle A B x. Through 

 the points r> and F draw the straight lines H K, H L, meeting 

 each other in the point H, and the straight line K L in the 

 points K and L. The triangle H K L circumscribed about the 

 circle D E F is equiangular to the given triangle ABC. 



PROBLEM XLIV. To inscribe a square in a given circle, and 

 about the same circle to circumscribe a square. 



Let A B c D (Fig. 63) be the given circle, and E its centre. 

 Through E draw the diameters A c, B D at right angles to each 

 other, and join A B, B c, c D, and D A. The figure A B c D thus 

 formed is a square, and it is inscribed in the given circle 

 A B c D, as required. 



To circumscribe a square about the circle A B c D, draw the 

 diameters AC, B D as before. Through the points A and c 

 draw the straight lines F G, H K parallel to B D, and through 

 the points B and D draw the straight lines F K, G H parallel 

 to A c. The figure F G H K thus formed is a square, and it 

 is circumscribed about the circle A B C D, as required. 



PROBLEM XLV. To inscribe a circle in a given square, and 

 about the same square to circumscribe a circle. 



Let F G H K (Fig. 63) be the given square : it is required to 

 inscribe a circle within the given square F G H K, and to circum- 

 scribe a circle about it. First bisect the sides F G, F K in 

 the points A and B, and through A draw A c parallel to F K or 

 o H, and through B draw B D parallel to F G or H K. From 



Fig. 62. 



the point E, the point of intersection of the straight lines A c, 

 B D, at the distance E A, E B, E c, or E D, describe the circle 

 A B c D. This circle touches the sides of the given square 

 F G H K, and is inscribed within it, as 

 required. 



To circumscribe a circle about the 

 given square F G H K, find the point 

 E as before, and then from the point 

 E as centre, with a radius equal to 

 the straight line joining E with any 

 one of the four corners of the square, 

 describe the circle F G H K. The 

 circumference of the circle F G H K 

 passes through the other three corners 

 of the square, and the circle F G H K 

 is therefore circumscribed about the 

 square F G H K, as required. 



We may now pass on to the construction of regular polygons. 

 The term polygon is derived from two Greek words, iroAus (pol-'use), 

 much or many, and ydivia. (go'nia), an angle, and means a figure 

 that has many angles. Many-angled figures are also called 

 multilateral or many-sided figures, from the Latin multus, much 

 or many, and latus, a side. " Polygon " and " multilateral 

 figure " are terms which may be considered to mean precisely 

 the same thing, for a figure that has many angles must, of 

 course, have many sides. It has, in fact, just so many sides as 

 it has angles, and the most familiar illustration of this that can 

 be given is that of a room, which, generally speaking, has just four 

 sides and four corners or angles. The terms " polygon " and 

 " multilateral figure " are applied, as we have been taught in 

 Definition 33 (Vol. I., p. 53), to any figure that has more than 

 four sides. A polygon may be regular or irregular that is to 

 say, its sides and angles may be equal or unequal. The student 

 has already been shown the method of making triangles equal 

 to given irregular polygons ; and the construction of an irregular 

 polygon of any number of sides, having its angles equal to 

 angles of any prescribed opening, would be a thing that he 

 could readily accomplish, provided that he has paid sufficient 

 attention to our lessons to understand thoroughly all that we 

 have advanced. It is with the construction of regular polygons 

 only that we have now to do. 



In Definition 34 we were further taught that polygons are 

 divided into classes according to the number of their sides and 

 angles. Some of these classes have no distinctive name, as will 

 be seen from the following table ; but many of them have a 

 name by which the number of their sides can be recognised at 

 once. Thus the polygon that has five sides and five angles is 

 called a pentagon, from the Greek Trfvre (pen'te), five, and yuvia, 

 an angle; the polygon that has six sides and six angles is called 

 a hexagon, from the Greek e| (hex), six, and ya>via, an angle ; 

 and so on, the Greek or Latin word for the number of the sides, 

 or some modification of it, being prefixed to the termination gon. 

 A triangle would be called a trigon, and a square a tetragon, 

 according to this system of naming figures from the number of 

 their angles. 



The number of degrees in the angle of any regular polygon 

 may be found arithmetically by the following process : 



The angles formed by any number of lines meeting together in 

 a point, such as the lines drawn from the angles of any polygon 

 to any point within it, or, in the case of a regular polygon, to 

 its centre, are together equal to four right angles, or 360 

 degrees. The greater the number of sides 

 of any regular polygon, the less will be 

 the angle at its centre, subtended by each 

 of its sides; and to find the number of 

 degrees contained in its opening, we have 

 to do nothing more than to divide 360 by 

 the number of sides. For example, in 

 the regular pentagon, or five-sided figure 

 A B c D E, in Fig. 64, it is clear that each 

 of the five angles, A F B, B F c, c F D, D F E, 

 E F A. formed by drawing straight lines 

 from its five salient angles at A, B, C, D B, 



to its centre, F, is equal to one-fifth of 360 degrees, or, in 

 other words, is an angle of 72 degrees. We now wish to 

 find the numerical value of any and all of the angles of the 

 polygon in degrees. We know that the three interior angles 

 of any triangle are together equal to two right angles, or 180 



Fig. 64. 



