Exercises.
1. The sum of the squares on the segments of a line of given length is a minimum when it is bisected.
2. Divide a given line internally, so that the sum of the squares on the parts may be equal to a given square, and state the limitation to its possibility.
3. If a line AB be bisected in C and divided unequally in D,
4. Twice the square on the line joining any point in the hypotenuse of a right-angled isosceles triangle to the vertex is equal to the sum of the squares on the segments of the hypotenuse.
5. If a line be divided into any number of parts, the continued product of all the parts is a maximum, and the sum of their squares is a minimum when all the parts are equal.
PROP. X.—Theorem.
If a line (AB) be bisected (at C) and divided externally (at D), the sum of the squares on the segments (AD, DB) made by the external point is equal to twice the square on half the line, and twice the square on the segment between the points of section.
