The First Six Books of the Elements of Euclid

by Euclid; John Casey

PROP. XVII.—Theorem

1. If three right lines (A, B, C) be proportional, the rectangle (A.C) contained by the extremes is equal to the square (B²) of the mean.
2. If the rectangle contained by the extremes of three right lines be equal to the square of the mean, the three lines are proportional.

Dem.—1. Assume a line D = B; then because A : B :: B : C, we have A : B :: D : C. Therefore [xvi.] AC = BD; but BD = B². Therefore AC = B²; that is, the rectangle contained by the extremes is equal to the square of the mean.

2. The same construction being made, since AC = B², we have A.C = B.D; therefore A : B :: D : C; but D = B. Hence A : B :: B : C; that is, the three lines are proportionals.

This Proposition may be inferred as a Cor. to the last, which is one of the fundamental Propositions in Mathematics.

Exercises.

1. If a line CD bisect the vertical angle C of any triangle ACB, its square added to the rectangle AD.DB contained by the segments of the base is equal to the rectangle contained by the sides.

Dem.—Describe a circle about the triangle, and produce CD to meet it in E; then it is easy to see that the triangles ACD, ECB are equiangular. Hence [iv.] AC : CD :: CE : CB; therefore AC.CB = CE.CD = CD² + CD.DE = CD² + AD.DB [III. xxxv.].