1. The squares on the diagonals of a quadrilateral are together double the sum of the squares on the lines joining the middle points of opposite sides. 2. If the medians of a triangle intersect in O, AB2 + BC2 + CA2 = 3(OA2 + OB2 + OC2). 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. 11. If perpendiculars be drawn from the angular points of a square to any line, the sum of the squares on the perpendiculars from one pair of opposite angles exceeds twice the rectangle of the perpendiculars from the other pair by the area of the square. 12. If the base AB of a triangle be divided in D, so that mAD = nBD, then 13. If the point D be taken in AB produced, so that mAD = nDB, then 14. Given the base of a triangle in magnitude and position, and the sum or the difference of m times the square on one side and n times the square on the other side, in magnitude, the locus of the vertex is a circle. 15. Any rectangle is equal to half the rectangle contained by the diagonals of squares described on its adjacent sides. 16. If A, B, C. &c., be any number of fixed points, and P a variable point, find the locus of P, if AP2 + BP2 + CP2+ &c., be given in magnitude. 17. If the area of a rectangle be given, its perimeter is a minimum when it is a square. 18. If a transversal cut in the points A, C, B three lines issuing from a point D, prove that 19. Upon the segments AC, CB of a line AB equilateral triangles are described: prove that if D, D′ be the centres of circles described about these triangles, 6DD′2 = AB2 + AC2 + CB2.