97. To three points fixing a triangle in either of two reciprocals must correspond also three rays fixing a triangle in the other reciprocal; hence, in general, triangle corresponds to triangle in reciprocals. But notice: the sides of one correspond to the vertices of the other; hence if the sides of one all go through the same point, the vertices of the other all lie on the same ray; that is, three concurrent rays in either reciprocal correspond to three collinear points in the other.

It now appears that axal and central symmetry are reciprocal to each other; the reciprocal of an axal symmetric is a central symmetric, and the reciprocal of a central symmetric is an axal symmetric; the reciprocal properties of axal symmetry are the properties of central symmetry, and the reciprocal properties of central symmetry are the properties of axal symmetry.

Very often the two symmetric figures may be regarded as the two halves of one figure; this one figure is then said to be symmetric as to the axis of symmetry or as to the centre of symmetry, as the case may be.

98. If our figure be two points, A and A', then the midnormal X of the tract AA' is the axis of symmetry, manifestly. If, now, any double point D on the axis be joined with A and A', there results the isosceles ▲ ADA', whence it appears that (Fig. 63)

The isosceles ▲ is a symmetric A.

It is plain that any two points on the ray AA' equidistant from N are symmetric as to X, that all points on the ray, and indeed in the whole plane, may be arranged in symmetric pairs, the members of each pair equidistant from the axis X.

99. Now take two points on the axis, as D and D', or D and D", and consider the 4-side DAD'A'. It is composed of two A, ADD' and A'DD', symmetric with each other as to the axis X, and opposed along that axis. Hence the 4-side is itself symmetrical as to X.

Def. Such a 4-side, with an axis of symmetry, is called a kite.

If we hold D fast, and let D' glide along X, the 4-side ADA'D' remains a kite. We see that there are two kinds

of kites, the convex kite, as ADA'D', and the re-entrant, as ADA'D". As the gliding point passes through N the kite changes from one kind to the other, passing through the intermediate form of the symmetrical A.

When the gliding point reaches a position D' such that ND ND', then the four sides of the kite are all equal (why?), and the kite becomes a rhombus (why?). In this case D and D' are symmetric as to AA' as an axis of sym

metry.

Hence the rhombus has two axes of symmetry; namely, its two diagonals.

In all cases the diagonals, AA' and DD', of the kite are normal to each other (why?).

100. Now consider a pair of points, B and B', symmetric as to the axis X (Fig. 64). Then X is mid-normal of BB'.

B

X

FIG. 64.

If C and C' be any other pair of symmetric points, then X is also mid-normal of CC'; hence BB' and CC' are parallel (why?). Also the tracts BC and B'C' are symmetric as to X (why?), and the 4-side BB'C'C is itself symmetric as to the axis X. Hence the angles at C and C' are equal,

also the angles at B and B' are equal (why?); hence the angles at B and C and at B' and C' are supplemental (why?), and the 4-side BB'C'C is an anti-parallelogram (why?). Hence we see that another symmetric 4-side is an anti-parallelogram.

It is plain that every anti-parallelogram is symmetric, for we know that the oblique sides prolonged yield an isosceles A. Let the student complete the proof.

101. There is only one kind of symmetric ▲, the isosceles. For, let ABA' (Fig. 65) be symmetric and A' correspondent

Α

B

*

FIG. 65.

to A. Then B must correspond to itself (why?); hence

B must lie on the axis (why?); hence BA = BA' (why?). Now let the student prove that

(1) In a symmetric ▲ the axis of symmetry is a medial; (2) it is also a mid-ray; (3) it is also a mid-normal.

Conversely, let him show that

A medial that is a mid-ray, or a mid-normal, is an axis of symmetry.

102. There are only two axally symmetric 4-sides; namely, the kite and the anti-parallelogram. For, in a symmetric 4-side a vertex must correspond to a vertex (why?). Also, not all vertices can be on the axis (why?). Also, a vertex on the axis is a double point (why?). Also, the vertices not on the axis must appear in pairs (why?); hence there must be either two or four of them. If there be two only, then the other two are on the axis and the 4-side is a kite; if there be four of them, we have just seen that the 4-side is an anti-parallelogram.

103. Now let us turn to the reciprocals. The reciprocals of the two points A and A' symmetric as to the axis X will be two rays L, L', symmetric as to the centre S. But rays symmetric as to a centre are parallel (why?); hence we have two parallels symmetric as to S, which is midway between them. The rays are symmetric as to any other point S' midway between them (why?). The piece of plane between these parallels is called a parallel strip, or band (Fig. 66).