The sum of these areas, or the area of the triangle ABC, (37.) If the sides of a triangle are 173.52 cm., 125.3 cm., and 96.357 cm., find the radius of the inscribed circle. PLANE GEOMETRY BOOK V REGULAR POLYGONS AND CIRCLES. SYMMETRY WITH RE SPECT TO A POINT 451. Defs.—A figure turns half-way round a point, if a straight line of the figure passing through the point turns through 180°, i. e., half of 360°. A figure turns one-third-way round a point, if a straight line of the figure passing through the point turns through 120°, i. e., one-third of 360°. In general, a figure turns one-nth way round a point if a straight line of the figure passing through the point turns through one-nth of 360°. 452. Exercise.—If a figure is turned half-way round on a point as a pivot, i. e., so that one straight line of the figure passing through that point turns through 180°, prove that every other straight line of the figure passing through that point turns through 180°. 453. Exercise.-In the same case, prove that every straight line not passing through the pivot makes after the rotation an angle of 180° with its original position. 454. Exercise.-If a figure turns one-third way round, prove that every straight line, whether passing through the pivot or not, makes after the rotation an angle of 120° with its original position. 455. Exercise.-If a figure turns one-nth way round, prove that every straight line of the figure makes after the rotation. 1 n an angle equal to of 360° with its original position. 456. Remark.-Hence we see the propriety of saying that when one straight line of the figure turns through an angle, the whole figure turns through the same angle. 457. Defs.-A figure was defined to be symmetrical with respect to a point, called the centre of symmetry (§ 40), if, on being turned half-way round on that point as a pivot, the figure coincides with its original position or impression. To distinguish this kind of symmetry from those which follow, it may be called two-fold symmetry with respect to a point. 458. Def.-A figure has three-fold symmetry with respect to a point, if, on being turned one-third way round on that point as a pivot, it coincides with its original impression. FIGURES POSSESSING THREE-FOLD SYMMETRY WITH RESPECT TO A POINT A figure which coincides with its original when turned one-third way round must also coincide when turned two-thirds. For, since it coincides after the first third, it may then be regarded as the original figure, and will therefore coincide when turned one-third again. When turned the third third the figure has completed one revolution, and each part is in its original position. It is easy to copy one of the above figures on tracing-paper or card-board, cut it out, fit it again to the page, stick a pin through its centre, and turn the figure one-third way round. In Propositions I. and II. it is convenient to think of the original diagram as fixed on the page, while another diagram, as the card-board, revolves upon it. 459. Defs.—We may define likewise four-fold, five-fold, etc., symmetry. In general a figure has n-fold symmetry with respect to a point, called the centre of symmetry, if, on being turned about that point one-nth of a revolution, it coincides with its original impression. Such a figure will also coincide if turned an ŋth of a revolution a second, third, fourth time, etc. For after the first th it becomes the original figure, and will therefore coincide when turned one-nth again. 460. Defs.-A triangle is regular, if it has three-fold symmetry with respect to a point. The point is called the centre of the triangle. A quadrilateral is regular, if it has four-fold symmetry; a pentagon if it has five-fold symmetry, etc. In general a polygon of n sides is regular, if it has n-fold symmetry. The centre of symmetry is called the centre of the polygon. A DOOO REGULAR TRIANGLE REGULAR REGULAR REGULAR REGULAR *This figure was used as a badge by the secret society founded by Pythago ras about 550 B.C. for the pursuit of Mathematics and Philosophy. supposed to possess mysterious properties, and was called Health." It was PROPOSITION I. THEOREM 461. Given a regular polygon: I. All its sides are equal. II. All its angles are equal. III. A circle may be circumscribed about it, its centre being the centre of the polygon. IV. A circle may be inscribed in it, its centre being the centre of the polygon. E FIG. I E B FIG. 2 GIVEN—ABCDE, a regular polygon of n sides with centre O. TO PROVE I. Its sides are equal. II. Its angles are equal. III. A circle can be circumscribed, with centre O. IV. A circle can be inscribed, with centre 0. I. (Fig. 1.) By definition, the polygon will, after being turned about O one-nth of a revolution, coincide with its original impression. § 460 Any side as AB must therefore take the position previously occupied by some other side. Since each turn is one-nth of a revolution, ʼn turns are necessary before AB resumes its original position. Hence in a complete revolution AB must coincide in succession with the n different sides of the polygon. Hence AB is equal to each of the other sides, and they are all equal to each other. Q. E. D. |