From C let fall the I CK;

and from D as a centre, with a radius equal to D B,

describe an arc cutting H B produced, at P.

verimeter

§ 113

Now

BK = } BH,
(a I drawn from the vertex of an isosceles A bisects the base),

PROPOSITION XXVII. THEOREM. 410. The maximum of isoperimetrical polygons of the same number of sides is equilateral.

B K

Let A B CD, etc., be the maximum of isoperimetrical

polygons of any given number of sides.
We are to prove A B, BC, C D, etc., equal.

Draw A C. The A ABC must be the maximum of all the A which are formed upon A C with a perimeter equal to that of A A BC.

Otherwise, a greater A A KC could be substituted for AA BC, without changing the perimeter of the polygon.

But this is inconsistent with the hypothesis that the polygon A B C D, etc., is the maximum polygon. .:: the A ABC, is isosceles,

§ 409 (of all s having the same base and equal perimeters, the isosceles A is the

maximum).
In like manner it may be proved that BC=C D, etc.

Q. E. D. 411. COROLLARY. The maximum of isoperimetrical polygons of the same number of sides is a regular polygon. For, it is equilateral,

§ 410 (the maximum of isoperimetrical polygons of the same number of sides is

equilateral).

Also it can be inscribed in a 0, $ 408 (the maximum of all polygons formed of given sides can be inscribed in a O). Hence it is regular,

§ 364 (an equilateral polygon inscribed in a О is regular).

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

B

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

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

The polygon 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. 6.

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 Q 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 Q is > 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.

418. Two points are symmetrical with re. spect 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.

PI

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

X

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

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 P P at M right angles.