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PROPOSITION XXII

217. Theorem. If a tangent and a secant are drawn from a point to a circumference, the tangent is a mean proportional between the secant and its external segment.

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PROPOSITION XXIII 220. Theorem. If two chords intersect within a circle, the product of the segments of one equals the product of the segments of the other. Dem. X = Y

D
A=D

E
A

B
A AEC

sim.
DEB
AE: CE = DE: BE
AE X BE= CE X DE

Y

х

PROPOSITION XXIV 221. Theorem. The product of two sides of a triangle equals the product of the diameter of the circumscribed circle and the perpendicular to the third side from the opposite vertex.

[blocks in formation]

=

66

Dem. ABD = rt. 2

[insc. in semi 0] C=D

[.“ same segment]
A ABD

sim.
AEC
AB: AD= AE: AC

AB X AC = AD X AE 222. Cor. The diameter of a circle circumscribed about a triangle equals what?

=

PROPOSITION XXV 223. Theorem. The product of two sides of a triangle equals the product of the segments of the third side formed by the bisector of the opposite angle plus the square of the bisector.

A

B

E

Appl.

Cons.

Prove AB X AC = BD X CD + AD2
Circumscribe O about A.
Produce AD to E. Draw EC

Dem.

mut. = - Z
A ABD
sim.

X = X' (hyp.)
AEC

B= E (insc. in same seg.)_
AB: AD = AE: AC
AB X AC AD X AE

AD (AD + DE)
= AD + AD X DE

AD

+ BD x CD ( prod. of seg. of AET

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EXERCISES

177. The sides of a triangle are 7, 8, and 12. The longest side of a similar triangle is 30. What are the other two sides ?

178. The bases of two similar triangles are 5 ft. 4 in. and 3 ft. 4 in. The altitude of the first is 4 ft. 10 in. What is the altitude of the second ?

REGULAR POLYGONS — DEFINITION

224. A Regular Polygon is one that is both equiangular and equilateral.

PROPOSITION XXVI

225. Theorem. Regular polygons of the same number of sides are similar.

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179. In Prop. XIX, if AC, BC, 23, and 14 are respectively 48, 36, 27, and 25, what are 13 and AD ?

180. Two isosceles triangles are similar if any angle of one equals the homologous angle of the other.

PROPOSITION XXVII

226. Theorem. A circle may be circumscribed about, or inscribed within, any regular polygon.

[blocks in formation]

Appl.

ABCDE = reg. O. Circles may be circumscribed and inscribed.

Cons.

Take O, the centre of a O pass

determined by ing through three consecutive

I bisectors of ED vertices E, D, and C.

and DC Join OB, OA, etc.

OCD = isos. A 1. X = Y

[°C A]

(= radii) A ODE

[ 2 s. and inc. _] OCB ОВ

Dem.

=

= OE

i.e. O, which passes through E, D, and C, also passes through B and (similarly] A, etc., and a circle is circumscribed about the regular polygon.

2. ED, DC, etc., are = distant chords of outer
from centre 0

O, = by hyp.
O described about ( as a centre,
with radius OF (I to ED),
will be an inscribed O

will touch sides of Lpolygon

of]

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