The First Six Books of the Elements of Euclid

by Euclid; John Casey

Exercises.

1. Show that Proposition vi. is reduced to Proposition v. by producing the line in the opposite direction.

2. Divide a given line externally, so that the rectangle contained by its segments may be equal to the square on a given line.

3. Given the difference of two lines and the rectangle contained by them; find the lines.

4. The rectangle contained by any two lines is equal to the square on half the sum, minus the square on half the difference.

5. Given the sum or the difference of two lines and the difference of their squares; find the lines.

6. If from the vertex C of an isosceles triangle a line CD be drawn to any point in the base produced, prove that CD² − CB² = AD.DB.

7. Give a common enunciation which will include Propositions v. and vi.

PROP. VII.—Theorem.

If a right line (AB) be divided into any two parts (at C), the sum of the squares on the whole line (AB) and either segment (CB) is equal to twice the rectangle (2AB.CB) contained by the whole line and that segment, together with the square on the other segment.