what could be made a parallelogram (142); hence ▲ A', BB', bisect each other in o (146).

727. SCHOLIUM. When two polygons are symmetrical with respect to a center, one can be made to coincide with the other by revolving it about the center through two right angles in their common plane.

728. DEFINITION. A figure is symmetrical with respect to a point, if every intercept that passes through the point is bisected there.

B

C

M

d

M'

II. SYMMETRY WITH RESPECT TO AN AXIS.

a

729. DEFINITION. Two points are said to be symmetrical

with respect to a straight line, if

this line bisects at right angles the straight line joining the two points.

Thus the points P and P' are

symmetrical with respect to X Y,

P

X

Y

if XY bisects PP'at right angles. The line XY is then called the axis of symmetry as regards P and P'.

EXERCISE 927. A circle is symmetrical with respect to what point?

928. A parallelogram is symmetrical with respect to what point?

929. A trapezium has no center of symmetry.

930. Every regular polygon of an even number of sides has a center of symmetry.

931. The axis of symmetry of the extremities of a chord is what line?

932. The axis of symmetry of opposite vertices of a square is what

730. DEFINITION. Any two figures are said to be symmetrical with respect to an axis, if

every point in the one has a

point in the other symmetrical x

C

with respect to that axis.

Thus the figures ABC, A'B'C',

are symmetrical with respect to

B

Y

Xy, if corresponding to every point in ABC there is a point in A'B'C' symmetrical with respect to XY.

731. SCHOLIUM. It is obvious that, if the portion of the plane above X Y be revolved about XY as an axis, till it coincides with the portion of the plane below X Y, the figure A B C will coincide with A'B'C', since the homologous points are at equal distances from X Y.

732. DEFINITION. A plane figure is symmetrical with respect to an axis, if the axis divides the figure into two symmetrical figures. Thus the figure AB ..... B' is symmetrical with regard to

X

XY if its homologous points are symmetrical with respect

to X Y.

EXERCISE 933. How many axes of symmetry may a circle have ? What common axis of symmetry have two circles?

934. An isosceles triangle is symmetrical with respect to which altitude?

B

935. An equilateral triangle has how many axes of symmetry ? 936. How many axes of symmetry may be drawn for (1) a square? (2) A rhombus ? (3) A regular pentagon? (4) A regular hexagon? (5) A regular polygon of 2n sides? (6) Of 2 n + 1 sides?

937. In a quadrilateral ABCD, AB=AD, and CB CD. Show that AC is an axis of symmetry, and is perpendicular to BD.

A

D

PROPOSITION II. THEOREM.

733. If a figure is symmetrical with respect to two axes at right angles to each other, it is also symmetrical with respect to their intersection as center.

To Prove: AB

H, symmetrical with respect to axes XX', YY',

...

H is symmetrical with respect to O as center.

From P, any point in the perimeter of the figure,

draw P Q R to YY', and through R draw RS P' to X X'.

Join OP, OP', and QS.

Since PQ QR,

and OS QR,

PQ OS.

Also P Q is to OS;

.. OP is and to QS.

In the same way it may be proved that

OP' is and = to QS;

(Hyp.)

(136)

(Ax. 1)

(106) (136)

.. POP' is a straight line, and is bisected in 0;

i.e., O is the center of symmetry of AB

...

H. Q.E.D.

734. SCHOLIUM. The axes XX', YY', evidently divide the figure into four equal parts. Any one of these parts may be made to coincide with either of the adjacent parts by revolving it about one of the axes, or may be made to coincide with the opposite part by revolving it, in the plane of the figure, through two right angles.

SYMMETRICAL POLYHEDRONS.

I. SYMMETRY WITH RESPECT TO A CENTER.

735. DEFINITION. Two polyhedrons are said to be symmetrical with respect to a center, when each vertex of the one has its symmetrical vertex on the other polyhedron.

Thus in the polyhedrons S-A B C, S'-A'B'C', if the lines joining the vertices A and A', B and B', etc., all pass through the same point 0, and are bisected in that point, the polyhedrons are said to be symmetrical with respect to 0, which is called their center of symmetry.

736. COR. If two polyhedrons are symmetrical with respect to a center, their homologous faces are severally equal, their dihedral angles are severally equal, their polyhedral angles are symmetrical, and the polyhedrons are equivalent.

737. DEFINITION. A polyhedron ABCD-A' is said to be symmetrical with respect to a center o, if its vertices, taken two and two, are symmetrical with

regard to 0; i.e., if A A', B B', etc., are each bisected in the same point 0.

738. COR. A polyhedron, in order

to have a center of symmetry, must have an even number of edges, must have its homologous edges equal and parallel,

D

A

B'

its homologous plane angles and dihedral angles equal, and its homologous polyhedral angles symmetrical.

II. SYMMETRY WITH RESPECT TO AN AXIS.

739. DEFINITION. Two polyhedrons are said to be symmetrical with respect to an axis, when this axis is an axis of symmetry for the corresponding vertices

of the two polyhedrons.

740. DEFINITION.

A polyhedron is

B

said to be symmetrical with respect to an axis, when this line is an axis of symmetry for the corresponding vertices of the polyhedron, taken two and two.

Thus the polyhedron AB... E' is symmetrical with respect to the axis oo'

B

when this axis is an axis of symmetry for each of the pairs of vertices, A and D, B' and E', etc.

III. SYMMETRY WITH RESPECT TO A PLANE.

741. DEFINITION. Two points are said to be symmetrical with respect to a plane, when the plane is perpendicular to, and bisects, the straight line joining the two points.

EXERCISE 938. Every intercept between two opposite faces of a polyhedron having a center of symmetry, and passing through that center, is bisected there.

939. Every prism whose bases are polygons symmetrical with respect to a point, has a center of symmetry.

940. Where is the center of symmetry of a parallelopiped ?

941. A right prism whose bases are symmetrical with respect to a center, has an axis of symmetry.

942. A rectangular parallelopiped has three axes of symmetry.

943. How many axes of symmetry has a cube?

944. A regular pyramid having an even number of lateral faces, has an axis of symmetry.