CHAPTER VIII.

SIMILARITY OF POLYGONS.

DEFINITIONS.

1. Similar Polygons are such as have the same form. Or, scientifically defined:

Similar Polygons are those which are mutually equiangular, and whose corresponding sides are proportional.

2. The points, lines, or angles similarly situated, in similar polygons, are said to be homologous.

H K, parallel to A B. The triangles A G H and D E F are equal throughout (Ch. II, Th. IV); hence, a = c, and G H = E F; but b = c; hence, a = b; hence, G H and B C are parallel; hence, AB AG :: AC: AH Ch. VII, Th. I, Cor. I.

But BK GH (Ch. II, Th. XXIII, Cor.), which, as we have seen, equals EF; hence, BK = E F. But AH is equal to D F;

(2) AC: DF:: BC: EF.

Combining (1) and (2),

AB: DE: AC: DF:: BC: EF.

Therefore, the triangles, being mutually equiangular, and having their corresponding sides proportional, are similar.

Q. E. D.

Cor. I.-Two triangles are similar, when they have two angles of the one equal to two angles of the other, each to each.

Cor. II. In similar triangles, the corresponding or homologous sides lie opposite equal angles.

This is evident from a simple inspection of the diagrams,— noting the relative position of the proportional sides.

Triangles which have their homologous sides proportional, are similar.

In the triangles A B C and DEF, let AB: DE: AC: DF BC: EF; then will they be similar.

For, take AG = DE, and

Ал

H

dC E

F

B

Hence, G H is parallel to B C (Ch. VII, Th. II, Cor.);

hence, a = b, and c = d;

hence, A G H and A B C are equiangular;

hence, AC AH:: BC: G H. But, by hypothesis,

Hence, the triangles A G H and D E F are equiangular (Ch. II, Th. XVII).

But A G H is equiangular to A B C; hence, D E F is equiangular to A B C.

Therefore, D E F and ABC are similar (Th. I). Q. E. D.

Scholium. It is thus seen that if triangles are mutually equiangular, they have their homologous sides proportional; and, conversely, if they have their homologous sides proportional, they are mutually equiangular. Either of these conditions involves the other. Hence, in establishing the similarity of two triangles, it is sufficient to prove, either that they are mutually equiangular, or that their homologous sides are proportional. To establish the similarity of other polygons, however, it will be necessary to prove both these conditions, since the equality of their angles does not involve the proB portionality of their sides, nor conversely. For example, EF being parallel to CD, the quadrilaterals

A

F

D

E

ABCD and ECDF are mutually equiangular, but the sides about these equal angles are not proportional; for if

AD: DF: CD: EF,

then, since CD < EF, AD would be less than D F, whereas it

is greater.

THEOREM III.

Triangles which have an angle in each equal, and the including sides proportional, are similar.

For, take A G = D E, and AHDF; draw G H, then, since AB DE AC: DF, it follows that

AB: AG: AC: AH; hence,

GH is parallel to B C; hence,

a = b, and c = d; hence, A G H is equiangular

to ABC; but DEF is equiangular to A G H, since A = D, AGDE, and AH = DF; hence, DEF is equiangular to A B C; hence, it is also similar to AB C.

ᅡ

THEOREM IV.

Q. E. D.

Triangles which have their sides parallel, each to each, are similar.

In the triangles A B C and DEF, let AB be parallel to DE, AC to DF, and BC to

EF; then will they be similar. B

For, since AB is

(Ch. II, Th. XI);

E4

G

parallel to DE, and B C to EF, B = E since BC is parallel to EF, and AC to

DF, CF; hence, AD; hence, the two triangles are equiangular; hence, they are similar.

Q. E. D.

THEOREM V.

Triangles which have their sides perpendicular, each to each, are similar.

B.

Q"

E

F

D

laterals Q, Q', Q'.

to four right angles

but

.'.

Q

In the triangles ABC and DEF, let DE be perpendicular to AB, DF to AC, and EF to BC; then will they be similar.

For, produce the sides of DEF till they meet the sides AC of AB C, forming the quadriThe four angles of Q are together equal (Ch. II, Th. XXVI, Cor.); but

r+r'two right angles;

e + C = two right angles;

be two right angles (Ch. II, Th. I); bee+C;

Therefore, two angles of the one triangle being equal to two angles of the other, each to each, the third angles are equal, and a B; hence A B C and D E F are equiangular; hence, they are also similar.

Q. E. D.

EXERCISES.

1. The sides of a triangle are 10, 15, and 20; the homologous sides of a second triangle are 5, 7, and 9; are they similar?

2. The sides of a triangle are 5, 7, and 9; the side of a similar triangle, homologous with 7, is 32: required the remaining sides of the latter.