The First Six Books of the Elements of Euclid

by Euclid; John Casey

In extending this proof to parallelograms we have only to use P instead of

P.

PROP. II.—Theorem.

If a line (DE) be parallel to a side (BC) of a triangle (ABC), it divides the remaining sides, measured from the opposite angle (A), proportionally; and, conversely, If two sides of a triangle, measured from an angle, be cut proportionally, the line joining the points of section is parallel to the third side.

1. It is required to prove that AD : DB :: AE : EC.

Dem.—Join BE, CD. The triangles BDE, CED are on the same base DE, and between the same parallels BC, DE. Hence [I. xxxvii.] they are equal, and therefore [V. vii.] the triangle ADE : BDE :: ADE : CDE;

	but	ADE	: BDE :: AD : DB [i.],
	and	ADE	: CDE :: AE : EC [i.].
	Hence	AD	: DB :: AE : EC.

2. If AD : DB :: AE : EC, it is required to prove that DE is parallel to BC.