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PROPOSITION XXVIII. THEOREM. 412. Of isoperimetrical regular polygons, that is greatest which has the greatest number of sides.

Let Q be a regular polygon of three sides, and Q be

a regular polygon of four sides, each having the same perimeter.

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The polygon Q may be considered an irregular polygon of four sides, in which the sides A D and D B make with each other an 2 equal to two rt. .

Then the irregular polygon Q, of four sides is less than the regular isoperimetrical polygon Q' of four sides,

§ 411 (the maximum of isoperimetrical polygons of the same number of sides is a

regular polygon).

In like manner it may be shown that Q is less than a regular isoperimetrical polygon of five sides, and so on.

Q. E. D.

413. COROLLARY. Of all isoperimetrical plane figures the circle is the maximum.

PROPOSITION XXIX. THEOREM. 414. If a regular polygon be constructed with a given area, its perimeter will be the less the greater the number of its sides.

Let Q and ll be regular polygons having the same

area, and let Q' have the greater number of sides. We are to prove the perimeter of Q > the perimeter of Q'.

Let Q" be a regular polygon having the same perimeter as Q', and the same number of sides as Q. Then Qis > Q",

§ 412 (of isoperimetrical regular polygons, that is the greatest which has the greatest

number of sides). But

Q=Q',

.::Q is > Q". .:. the perimeter of Q is > the perimeter of Q". But the perimeter of Q= the perimeter of Q", Cons. .: the perimeter of Q is > that of Q'.

Q. E. D.

415. COROLLARY. The circumference of a circle is less than the perimeter of any other plane figure of equal area.

ON SYMMETRY. — SUPPLEMENTARY. 416. Two points are Symmetrical when they are situated on opposite sides of, and at equal distances from, a fixed point, line, or plane, taken as an object of reference.

417. When a point is taken as an object of reference, it is called the Centre of Symmetry; when a line is taken, it is called the Axis of Symmetry; when a plane is taken, it is called the Plane of Symmetry.

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418. Two points are symmetrical with respect to a centre, if the centre bisect the straight line terminated by these points. Thus, P, P' are symmetrical with respect to C, if C bisect the straight line P P.

419. The distance of either of the two symmetrical points from the centre of symmetry is called the Radius of Symmetry. Thus either C P or C P' is the radius of symmetry.

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420. Two points are symmetrical with respect to an axis, if the axis bisect at right angles the straight line terminated by these points. Thus, P, P' are symmetrical with respect to the axis X X', if X X' bisect P Plat right angles.

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421. Two points are symmetrical with respect to a plane, if the plane bisect at right angles the straight line terminated by these points. Thus P, P are symmetrical with respect to M N, if M N bisect PP' at right angles.

422. Two plane figures are symmetrical with respect to a centre, an axis, or a plane, if every point of either figure have its corresponding symmetrical point in the other.

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Α' Α'
Fig. 1.
Fig. 2.

Fig. 3. Thus, the lines A B and A' B' are symmetrical with respect to the centre C (Fig. 1), to the axis X X' (Fig. 2), to the plane MN (Fig. 3), if every point of either have its corresponding symmetrical point in the other.

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Also, the triangles A B D and A' B' D' are symmetrical with respect to the centre C (Fig. 4), to the axis X X' (Fig. 5), to the plane M N (Fig. 6), if every point in the perimeter of either have its corresponding symmetrical point in the perimeter of the other.

423. Def. In two symmetrical figures the corresponding symmetrical points and lines are called homologous.

Two symmetrical figures with respect to a centre can be brought into coincidence by revolving one of them in its own plane about the centre, every radius of symmetry revolving through two right angles at the same time.

Two symmetrical figures with respect to an axis can be brought into coincidence by the revolution of either about the axis until it comes into the plane of the other.

424. DEF. A single figure is a symmetrical figure, either when it can be divided by an axis, or plane, into two figures symmetrical with respect to that axis or plane; or, when it has a centre such that every straight line drawn through it cuts the perimeter of the figure in two points which are symmetrical with respect to that centre.

C

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Thus, Fig. 1 is a symmetrical figure with respect to the axis X X', if divided by XX' into figures A B C D and A B'C'D which are symmetrical with respect to X X'.

And, Fig. 2 is a symmetrical figure with respect to the centre 0, if the centre o bisect every straight line drawn through it and terminated by the perimeter.

Every such straight line is called a diameter.

The circle is an illustration of a single figure symmetrical with respect to its centre as the centre of symmetry, or to any diameter as the axis of symmetry.

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