Page images
PDF
EPUB

PROPOSITION XIII

202. Theorem. The bisector of an exterior angle of a triangle divides the opposite side externally into segments proportional to the adjacent sides.

[blocks in formation]

203. Sch. If a line is divided so that the ratio between its internal segments equals the ratio between its external segments, the line is said to be divided harmonically.

The bisector of an angle of a triangle and the bisector of the exterior angle at the same vertex divide the opposite side of the triangle harmonically, because the ratios between the segments of this side, both internal and external, equal the ratio between the adjacent sides.

SIMILAR POLYGONS

204. Definition. Two polygons are similar when they are mutually equiangular and have their homologous sides proportional.

B

[blocks in formation]

Fig. 1.

M

A

23

[ocr errors]

54

Fig. 2.

B
M and N (Fig. 1) are
similar if
A=1, B=2, C=3, etc.,

AB BC CD and

etc. 12

34' M and N (Fig. 2) are

N

с not similar, even if A=1, B= 2, etc.

AE ED unless

which 15

E

D is evidently not true.

M and N (Fig. 3) are not similar, even if the homologous sides are propor

B tional, because the homologous

2x
angles are not
equal.

A
M

1 205. Similar

N
2 v
figures have the
same form

20
D

5

4 shape. The ratio between homologous sides is the ratio of similitude of the figures.

2 Y

C

X

Y

3

22

[ocr errors]

z

or

U

FIG. 3.

PROPOSITION XIV

206. Theorem. Two triangles are similar if they are mutually equiangular.

[merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

170. In a triangle ABC, AB = 6, BC = 10, and CA = 12. Find the segments of CA, formed by a bisector of the angle B.

PROPOSITION XV 207. Theorem. Two triangles are similar if their homologous sides are proportional.

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
[merged small][merged small][ocr errors][merged small][merged small]

=

DE || BC
X= B
A ABC

sim.
ADE
AB BC
AD DE
AB BC
12 23
DE=23
A ADE

123
A ABC

sim. 123

[merged small][merged small][ocr errors][subsumed][merged small]

PROPOSITION XVI

208. Theorem. Two triangles are similar if an angle of one equals an angle of the other and the including sides are proportional.

[blocks in formation]
« PreviousContinue »