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