THEOREM VI.

Of all regular isoperimetric polygons, that is the greatest which has the greatest number of sides.

Let DE be the half side of one of the polygons, O its center, OE its apothegm. AB, C, CB the same for the other; DOE, ACB will be the half angles at the centers of the polygons; and as these angles are not equal, the lines CA, OD, prolonged, will meet at the point F; from this point draw FG perpendicular to OC; with O and C as centers, describe arcs GI, GH.

Now, DE is to the perimeter of the first polygon as O is to four right angles, and AB: perim. 2d polyg.:: C: 4 r. angs... DE: AB:: 0:0, GI GH and .. (th. 71, corol. 3), DE: AB::

0:

CG

cedents by OG, and the consequents by CG,

Multiplying the ante

DEX OG: AB × CG:: GI: GH.

But the similar triangles ODE, OFG give

OE: OG :: DE: FG... OE X FG = DE X OG (th. 54).

In the same manner it may be shown that

AB X CGCB X FG

Therefore, by substitution,

OE X FG: CB X FG: GI: GH.

If, then, it can be shown that the arc GI > arc GH, it will follow that the apothegm OE is greater than the apothegm CB.

Make the figures CKx = CGx, CKH = CGH.

Then KG > KHG (see exercise 13 of the miscellaneous exercises).

Much more

• Gx=} KxG >GH=}KHG.

GI>GH.

Q. E. D.*

Corollary from the preceding Propositions,―The circle is the greatest of all figures of the same perimeter; for it may be regarded as a regular polygon of an infinite number of sides.

*It has been seen in the note to corol. 4, th. 16, that equilateral triangles, squares, and regular hexagons are the only figures which will, in juxtaposition, leave no intervening space. It appears, also, from the present proposition, that the space inclosed in the regular hexagon is greater than that inclosed in the square or triangle of the same perimeter. Some writers on natural theology call attention to the fact that the cells of the bee-hive being made in the form of regular hexagons, thus affording the greatest space with a given amount of the material employed in their construction, indicate an instinct working in accordance with the most recondite principles of geometry.

APPENDIX II.

CENTERS OF SYMMETRY.

Def. 1. When the vertices of two polygons, or of the same polygon, are two and two upon lines meeting in a point interior, and at equal distances from this point, the point is called the center of symmetry.

Theorem 1. Prove that all lines drawn to opposite parts of the figure through the center of symmetry are equally divided at this point. 2. That the opposite sides of the figure or figures are equal, parallel, and arranged in a reverse order. 3. The converse.

Def. 2. Two points are said be situated symmetrically with respect to a line when this line is perpendicular to that which joins the two points, and divides it into two equal parts.

OF AXES OF SYMMETRY.

Two polygons, or portions of one polygon, are said to be symmetrical with respect to a line when their corresponding vertices are symmetrical. The line in such a case is called an axis of symmetry. Def. 3. An isosceles trapezoid is one whose inclined sides are equal.

Theorem. Prove that the line joining the middle points of the parallel bases of such a figure is an axis of symmetry.

Theorem 2. Prove that the isosceles trapezoid may be inscribed in a circle.

General Theorem. Prove that every figure which has two axes of symmetry perpendicular to each other has a center of symmetry at their intersection.

Schol. Show that these axes divide the figure into four equal parts.

OF DIAMETERS.

Theorem 3. When the vertices of two polygons or of a same polygon are two and two upon lines parallel, and equally divided by a MEDIAN line, this median line bisects, also, every other line parallel to the former, and terminating at the sides of the figure or figures.

CENTER OF MEAN DISTANCES.

Def. 4. If the middle points of the consecutive sides of a polygon be joined, a new polygon will be formed of less perimeter and area,

evidently, than the perimeter and area of the first. Proceeding in the same manner with the second, a third is obtained, still smaller, and so on. These operations being continued indefinitely, the result will be at length a polygon, infinitely small, which may be regarded as a point. This point is called the center of mean distances. It has a remarkable property. The distance of this point from any given line is equal to the quotient of the sum of the distances of all the vertices of the polygon from this given line, divided by the number of vertices. The student may prove this by proving the sum of the distances of the vertices of the second polygon equal to that of the first, and so on, till the polygons are reduced to a point, the center of mean distances.

Cor. From this will follow a construction for determining the center of mean distances, viz., determine its distance from two given lines by dividing the sum of the distances by the number of vertices of the polygon, and, drawing parallels to these two lines at the distances thus determined, these parallels will, by their intersection, determine the point required.

Def. 5. Polygons are said to be inversely similar when one is similar to a polygon symmetric with the other.

CENTERS OF SIMILITUDE.

Def. 6. The center of similitude is a point placed in such a manner with reference to two polygons, directly similar, as that, if a line be drawn through it to two homologous vertices of the polygons, the direction of these vertices from the point shall be the same, and the lines proportional to the homologous sides.

The distances from this point to the homologous vertices are called radii of similitude.

If, from a point taken at pleasure, lines be drawn to the vertices of a polygon, and upon these lines or their prolongations parts be taken proportional to them, the points thus obtained will determine a new polygon similar to the given polygon, and the arbitrary point will be the center of similitude of the two polygons.

The center of similitude may be either external or internal to the two polygons. (See diagram of note to th. 69 for an external center.) Two similar polygons which have their sides respectively parallel, and directed the same or contrary ways, have in the first case an external, and in the second case an internal center of similitude. In the latter case it is between homologous vertices.

Theorem. Prove that when three similar polygons have their sides respectively parallel, their three centers of similitudes are upon the same line.

THEOREM.

Two polygons (directly) similar, situated in any manner upon a plane, have always a common homologous point.*

By this is to be understood that there exists in the plane of the two polygons a point such that, if it be joined with the vertices of the two polygons, the homologous lines of junction will have the ratio of similitude of the two polygons,† and that the angles formed by these lines are equal each to each.

E

D

a

C

N

Let ABCDE, abcde be the two polygons, and N the point in which the two sides AB, ab meet. Find P, the center of a circle passing through A, a and N, or equally distant from these three points, and Q a point equally distant from B, b and N. Join PQ. Then find the point O symmetric to N, with reference to the line PA. O will be the point required.

* That is, a point from which homologous lines, drawn to the vertices of the two polygons, will have the ratio of similitude of the polygons, and form with each other equal angles.

This theorem is due to M. Chasles, and is demonstrated in the Bulletin des Sciences Mathématique of Férussac for 1830.

+ See note to def. 67.