H B! G E of it, BE, ED are equal to one another, it is a square, and what was required is now done : But if they are not equal, produce one of them BE to F, and make EF equal to ED, and bisect BF in G; and from the centre G, at the distance GB, or GF, describe the semicircle BHF, and produce DE to H, and join GH. Then, because the straight line BF is divided into two equal parts at G and into two unequal at E, the rectangle BE, EF, together with the square of GE, is equal to the square of GF (2. 5): But GF is equal to GH; therefore the rectangle BE, EF, together with the square of GE, is equal to the square of GH: But the square of GH is equal to the squares of GE, EH ; therefore the rectangle BE, EF, together with the square of GE, is equal to the squares of GE, EH: Take away the square of GE, which is common to both ; and the remainder, the rectangle BE, EF, is equal to the remainder, the square of EH: But the rectangle contained by BE, EF is the parallelogram BD, because EF is equal to ED; therefore BD is equal to the square of EH: But BD is equal to the rectilineal figure A; therefore the square of EH is equal to the rectilineal figure A: Wherefore a square has been made equal to the given rectilineal figure A, viz, the square described upon EH. Q. E, F. H PROBLEMS. BOOK II. 1. If a line be drawn from one of the acute angles of a right-angled triangle to the bisection of the opposite side, the square upon that line is less than the square upon the hypothenuse by three times the square upon half the line bisected. 2. If from the middle point of one of the sides of a right-angled triangle a perpendicular be drawn to the hypothenuse, the difference of the squares of the segments so formed is equal to the square of the other side. 3. In any triangle, if a perpendicular be drawn from the vertex to the base, the difference of the squares upon the sides is equal to the difference of the squares upon the segments of the base. 4. Let AOB be a quadrant of a circle, whose centre is 0; from any point C in its arc draw. CD perpendicular to OA or OB, meeting in E the radius which bisects the angle AOB : then shew that the squares upon CD, DE are together equal to the square upon OA. 5. If from any point in the diameter of a semicircle two lines be drawn to the circumference, one to the bisection of the arc, and the other perpendicular to the diameter, then the squares upon these two lines are together double of the square upon the radius. 6. If A be the vertex of an isosceles triangle ABC, and CD be drawn perpendicular to AB, prove that the squares upon the three sides are together equal to the square on BD and twice the square on AD and thrice the square on CD. 7. If from any point perpendiculars be dropped on all the sides of any rectilineal figure, the sum of the squares upon the alternate segments of the sides will be equal. 8. If from one of the acute angles of a right-angled triangle a line be drawn to the opposite side, the squares of that side and the line so drawn are together equal to the squares of the hypothenuse and the segment adjacent to the right angle. 9. Describe a square equal to the difference of two given squares. 10. Divide, when possible, a given line into two parts, so that the sum of their squares may be equal to à given square. 11. From D the middle point of AC, one of the sides of an equilateral triangle ABC, draw DE perpendicular on BC; and shew that the square upon BD is threefourths of the square upon BC, and the line BE threefourths of BC. 12. If from the vertex A, of a right-angled triangle BAC, AD be dropped perpendicular on the base, shew that the rectangles of BC and BD, BC and CD, BD and CD, are respectively equal to the squares upon AB, AC, AD. 13. Produce a given line so that the rectangle of the whole line produced and the original line shall be equal to a given square. 14. If on the radius of a circle a semicircle be described, and a perpendicular to the common diameter be drawn, the square of the chord of the greater circle, between the extremity of the diameter and the point of section of the perpendicular, will be double of the square of the corresponding chord of the lesser circle. 15. Divide a line in two points equally distant from its extremities, so that the square on the middle part shall be equal to the sum of the squares on the extremes : and shew also that in this case the square of the whole line will be equal to the squares of the extreme parts together with twice the rectangle of the whole and the middle part. 16. Divide a line into two parts, so that the squares of the whole line and one of the parts shall be together double of the square of the other part : and shew that, by the same division, the square of the greater part will be equal to twice the rectangle of the whole and the lesser part. 17. Divide a straight line into two parts so that the sum of their squares may be the least possible. 18. Shew that the sum of the squares upon two lines is never less than twice their rectangle, and that the difference of their squares is equal to the rectangle of their sum and difference. 19. Shew that of the two algebraical expressions, (a+x) (a—x)+x?=a’, (a +x)2+(a-x)?=2a2+2x2, the first is equivalent to Props. v. and vi., and the second to Props. Ix, and x., of Euc. 2. 20. ABCD is a rectangle, E any point in BC, F in CD: shew that the rectangle ABCD is equal to twice the triangle AEF together with the rectangle BE, DF. 21. If a line be divided into two equal and also into two unequal parts, the squares of the two unequal parts. are together equal to twice the rectangle contained by these parts together with four times the square of the line between the points of section. 22. If from one of the equal angles of an isosceles triangle a perpendicular be dropped on the opposite side, the rectangle of that side and the segment of it between the perpendieular and base is equal to half the square upon the base. 23. A, B, C, D, are four points in the same line, E a point in that line equally distant from the middle of the segments AB, CD, Ė any other point in AD : shew that the squares of AF, BF, CF, DF, are together greater than the squares of AE, BE, CE, DE, by four times the square of EF, 24. If from the extremities of any chord in a circle lines be drawn to any point in the diameter to which it is parallel, the sum of their squares is equal to the sum of the squares upon the segments of the diameter. 25. If the sides of a triangle be as 2, 4, 5, shew whether it will be acute- or obtuse-angled. 26. In any isosceles triangle ABC, if AD be drawn from the vertex to any point in the base, shew that the difference of the squares on AB and AD is equal to the rectangle of BD and CD. 27. If in the figure, Euc. 1. 47, the angular points be joined, the sum of the squares of the six sides of the figure so formed is equal to eight times the square of the hypothenuse. 28. If one angle of a triangle be four-thirds of a right angle, the square of the side subtending that angle is equal to the sum of the squares of the sides containing it, together with the rectangle contained by these sides. 29. If ABC be a triangle, with the angles at B, C, each double of the angle at A, then the square of AB is equal to the square of BC, together with the rectangle of AB and BC. 30. In any triangle ABC, if BP, CQ be drawn perpendicular to AC, AB, produced if necessary, then shall the square of BC be equal to the rectangle of AB, BQ, together with the rectangle of AC, CP. 31. In [22] shew that the square of the perpendicular is equal to the square of the line between the perpendicular and the other equal angle, together with twice the rectangle of the segments of the side. 32. If from the right angle of a right-angled triangle lines be drawn to the opposite angles of the square described on the hypothenuse, the difference of the squares on these lines is equal to the difference of the squares on the two sides of the triangle. 33. In any triangle the squares of the two sides are together double of the squares of half the base, and of the line joining its middle point with the opposite angle. |