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PROPOSITION VIII

195. Theorem. In a continued proportion, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent.

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196. Theorem. If a series of parallels intercept equal distances on one of two transverse lines they intercept also equal distances on the other.

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PROPOSITION X

197. Theorem. A line parallel to one side of a triangle divides the other two sides proportionally.

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Appl.

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When the side AB and its segment are commensurable.

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Cons.

AE

Apply AF, a common measure, to AB and AD, and let it be contained in AB m times, and in AC n times.

Through points of equal division draw parallels to

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Appl.

Cons.

Case II (Fig. 2)

When AB and AD are incommensurable

Apply some measure of AB to AD, contained in AD a certain number of times, to the point F, with remainder FD, < measure.

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and in general, if a line, or if any number of lines, be drawn parallel to a side of a triangle, the corresponding segments of the other two sides are proportional.

NOTE. - Hereafter, in the demonstration of theorems, the minor reasons for the principal or important steps of proof will often be omitted, and the student is expected to supply the reasons, as required.

PROPOSITION XI

199. Theorem. Conversely, a line which divides two sides of a triangle proportionally is parallel to the third side.

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169. If two parallels to AC, a side of a triangle ABC, meet the side AB at D and F, and the side BC at E and G, respectively, prove

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PROPOSITION XII

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

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201. Sch. By the segments of a line are meant the distances from a point of division on the line to the extremities of the line. The point of division may be on the line or on the line produced, forming respectively internal and external segments. In the former case the line equals the sum of its segments, and in the latter the difference.

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