18. Prove that a side and the diagonal of a square are incommensurable.
19. The diagonals of a lozenge are 16 and 30 metres respectively; find the length of a side.
20. The diagonal of a rectangle is 4.25 perches, and its area is 7.50 square perches; what are its dimensions?
21. The three sides of a triangle are 8, 11, 15; prove that it has an obtuse angle.
22. The sides of a triangle are 13, 14, 15; find the lengths of its medians; also the lengths of its perpendiculars, and prove that all its angles are acute.
23. If the sides of a triangle be expressed by m2 + n2, m2 − n2, and 2mn linear units, respectively; prove that it is right-angled.
24. If on each side of a square containing 5.29 square perches we measure from the corners respectively a distance of 1.5 linear perches; find the area of the square formed by joining the points thus found.
Exercises on Book II.
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).