3. Through a given point O draw three lines OA, OB, OC of given lengths, such that their extremities may be collinear, and that AB = BC.
4. If in any quadrilateral two opposite sides be bisected, the sum of the squares on the other two sides, together with the sum of the squares on the diagonals, is equal to the sum of the squares on the bisected sides, together with four times the square on the line joining the points of bisection.
5. If squares be described on the sides of any triangle, the sum of the squares on the lines joining the adjacent corners is equal to three times the sum of the squares on the sides of the triangle.
6. Divide a given line into two parts, so that the rectangle contained by the whole and one segment may be equal to any multiple of the square on the other segment.
7. If P be any point in the diameter AB of a semicircle, and CD any parallel chord, then
8. If A, B, C, D be four collinear points taken in order,
9. Three times the sum of the squares on the sides of any pentagon exceeds the sum of the squares on its diagonals, by four times the sum of the squares on the lines joining the middle points of the diagonals.
10. In any triangle, three times the sum of the squares on the sides is equal to four times the sum of the squares on the medians.