each other, the squares of its diagonals are together equal to the squares of its two sides, which are not parallel, and twice the rectangle contained by its parallel sides. 15. The sum of the perpendiculars let fall from any point within an equilateral triangle, upon the three sides, is equal to the perpendicular let fall from one of the angles upon the opposite side. 16. The squares of the diagonals of a trapezium are together double the squares of the two lines joining the bisections of the opposite sides. 17. The squares of the diagonals of a trapezium are together less than the squares of the four sides, by four times the square of the line joining the points of bisection of the diagonals. 18. If from any point within (or without) a trapezium perpendiculars be let fall on every side, the sum of the squares of the alternate segments made by them will be equal. 19. To divide a given line into two parts, such that the squares described upon them shall be equal to a given square. Show when the problem be comes impossible. 20. Divide a given straight line into two parts, such that twice the rectangle contained by them may be equal to the square of one of the parts. 21. Divide a given straight line into two parts, such that nine times the square described upon one part may be equal to the square described upon the other. 22. Divide a given straight line into two parts, such that the square of the whole line may be equal to the square of one of the parts, together with four times the rectangle contained by the two parts. 23. Divide a given straight line into three parts, such that the square upon the sum of the greatest and least parts may be four times the square upon the remaining part; and the sum of the squares upon the two least parts may be equal to the square upon the greatest. 24. Upon a given line, as an hypotenuse, to describe a right-angled triangle, such that the hypotenuse, together with the less of the two sides, shall be double the greater. 25. Show that the algebraical proposition (a+b)2+(ab)2=2a2+2b2. is equivalent to Props. 9 and 10 of the Second Book of Euclid. 26. Find algebraical propositions equivalent to Propositions 1, 2, 3, 4, 5, 6, 7, and 8 of the Second Book of Euclid. EXERCISES ON BOOK III. PROBLEMS. 1. To describe a circle which shall touch a given straight line in a given point, and also touch a given circle. Let FK be the given circle, and c the given point in the straight line XY: It is required to describe a circle which shall touch XY in the point c, and also touch the circle F K. Through c draw (E. I. 11.) the line DCB perpendicular to XY; find (E. III. 1.) the centre a of the circle FK, and A draw any radius AF; make CD equal to AF, and join A, D; from a draw (E. 1. 23.) AB, making the angle DAB equal to the angle ADE, and let DCB meet AB in the point B. Then B is the centre of the circle required. Since the angle DAB is equal to the angle ADB (constr.), therefore (E. I. 6.) BD is equal to BA; and CD is equal to AF or AK; therefore the remainder BC is equal to the remainder BK, and consequently a circle described from в as a centre, with the radius BC, will pass through K, and (E. I. 16.) it will touch XY in C, and moreover it will touch the circle FK in K, for a line drawn perpendicular at K will form a common tangent to the circles FK and CK. D F 2. Given the base, the vertical angle, and the perpendicular height of a triangle, to construct it. Let AB be the given base: Upon Aв describe (E. 11. 33.) a segment of a circle ADB, which shall contain an angle equal to the given vertical angle; from в draw BF perpendicular to A B, and make BF equal to the given A height of the triangle; through F E S B draw FD parallel to AB, meeting the arc of the circle in D; join B, D and A, D: Then ABD is the triangle required. Draw DS parallel to FB, then BFDs is a parallelogram, and (E. I. 34.) DS is equal to FB; and (constr.) because the angle SBF is a right angle, therefore (E. 1. 29.) the angle DSA is a right angle, that is, DS is perpendicular to AB, and it has been shown to be equal to FB, which was taken equal to the given perpendicular; moreover (constr.) the angle ADB is equal to the given vertical angle; therefore ADB is the triangle which was to be constructed. 3. To determine the position of a point of observation, D, at which lines drawn from three objects (A, B, C), whose distances from each other are given, shall make with each other angles equal to given angles; that is, let the angles ADB and ADC be given angles. Let A, B, C be the three given points or objects; join AB, and on it describe (E. III. 33.) a segment of a circle ADB, which shall contain an angle equal to the angle which the lines drawn from A and B are to include. Describe the whole circle ADBE, and make (E. 1. 23.) the angle ABE equal to the angle which the lines drawn from A and care to include. Join C, E; and 南B produce it to D, the circumference of the circle ADBE; then D is the position of the point required. Join A, E; then (E. II. 21.) the angle ADC is equal to the angle ABE, which was made equal to the angle formed by the lines drawn from a and c; and, moreover, the angle ADB is, by construction, equal to the angle formed by the lines drawn from A and B. 4. Through a given point within a circle, to draw a chord which shall be bisected in that point. 5. To draw a tangent to a circle, which shall be parallel to a given straight line. 6. Given the hypotenuse and the perpendicular let fall upon it from the opposite angle, to construct the right-angled triangle. 7. Divide the circumference of a circle into six equal parts. 8. From two given points on the same side of a line given in position, to draw two straight lines which shall contain a given angle, and be terminated in that line. 9. To describe a circle which shall have a given radius and its centre in a given line, and shall also touch another line, inclined to the former at a given angle. 10. To describe a circle which shall pass through a given point, and touch a given line in another given point. 11. To describe a circle which shall pass through a given point, and touch a given circle in another given point. 12. To describe a circle which shall have a given radius, and which shall touch two given lines, not parallel to each other. 13. To describe a circle, which shall touch a straight line in a given point, and also touch a given straight line. 14. To describe two circles, each having a given radius, which shall touch each other, and the same given straight line on the same side of it. 15. To describe three circles of equal diameters, which shall touch each other. 16. To describe a circle which shall pass through a given point, have a given radius, and touch a given straight line. 17. To describe a circle which shall pass through two given points, and touch a given straight line. 18. To describe a circle the centre of which may be in the perpendicular of a given right-angled triangle, and the circumference pass through the right angle and touch the hypotenuse. 19. Through a given point to draw a straight line, the part of which intercepted by the circle, |