The First Six Books of the Elements of Euclid

by Euclid; John Casey

Dem.—Let the same construction be made;

	then		AD : DB :: the triangle ADE : BDE [i.].
	and		AE : EC :: the triangle ADE : CDE [i.];
	but		AD : DB :: AE : EC (hyp.).
	Hence		ADE : BDE :: ADE : CDE.

Therefore [V. ix.] the triangle BDE is equal to CDE, and they are on the same base DE, and on the same side of it; hence they are between the same parallels [I. xxxix.]. Therefore DE is parallel to BC.

Observation.—The line DE may cut the sides AB, AC produced through B, C, or through the angle A; but evidently a separate figure for each of these cases is unnecessary.

Exercise.

If two lines be cut by three or more parallels, the intercepts on one are proportional to the corresponding intercepts on the other.

PROP. III.—Theorem.

If a line (AD) bisect any angle (A) of a triangle (ABC), it divides the opposite side (BC) into segments proportional to the adjacent sides. Conversely, If the segments (BD, DC) into which a line (AD) drawn from any angle (A) of a triangle divides the opposite side be proportional to the adjacent sides, that line bisects the angle (A).

Dem.—1. Through C draw CE parallel to AD, to meet BA produced in E. Because BA meets the parallels AD, EC, the angle BAD [I. xxix.] is equal to AEC; and because AC meets the parallels AD, EC, the angle DAC is equal to ACE; but the angle BAD is equal to DAC (hyp.); therefore the angle ACE is equal to AEC; therefore AE is equal to AC [I. vi.]. Again, because AD is parallel to EC, one of the sides of the triangle BEC, BD : DC :: BA : AE [ii.]; but AE has been proved equal to AC. Therefore BD : DC :: BA : AC.