Secondly, to divide it externally. HEA Construction. Produce BA, and take AH equal to AB. Bisect AH in E. On AB describe a square ACDB. Join EC; from EH produced, cut off EK= EC. Then will AB. BK= AK?. . Proof. On KA describe a square KGLA, and produce DC to meet KG in F. Then, since AH is bisected in E and produced to K, AK ~ KH= EK’- EA; (11. 8.) but EK’= EC", and therefore EK- EA = AC, therefore ARX KH=AC; that is, the figure FL= the figure AD. Add to each KC, then the figure KL = the figure KD; that is, the square on AK is equal to the rectangle ABxBK. Q. E. D. EXERCISES. 1. Construct a square double of a given square. 2. Construct a square equal to two, or three, or any number of given squares. 3. Divide a straight line into two parts, such that the square of one of the parts may be half the square on the whole line. 4. Given the base, area, and one of the angles at the base, construct the triangle. 5. Find the locus of a point which moves so that the sum of the squares of its distances from two given points is constant. We subjoin a few problems and theorems as miscellaneous exercises in the Geometry of angles, lines, triangles, parallelograms, and the equality of areas. MISCELLANEOUS THEOREMS AND PROBLEMS. 1. Prove that the acute angle between the bisectors of the angles at the base of an isosceles triangle is equal to one of the angles at the base of the triangle. 2. Find a point equally distant from three given straight lines. 3. If the diagonals of a quadrilateral bisect one another and are equal to one another, the figure will be a rectangle. 4. If the diagonals of a quadrilateral bisect one another at right angles and are also equal, the figure will be a square. 5. If ABC is a triangle, AB being greater than AC, and a point D in AB be taken such that AD= AC; prove that the angle BCD is equal to half the difference of the angles ABC, ACB. 6. If ABCD is a parallelogram, and AE= CF are cut off from the diagonal AC, then BEDF will be a parallelogram. 7. If AA' = CC' be cut off from the diagonal AC, and BB' = DD' from the diagonal BD of a parallelogram, then will A'B'C'D' be also a parallelogram. 8. If AA'= BB' = CC' = DD be cut off from the sides of the parallelogram ABCD taken in order, then will A'B'C'D' be also a parallelogram. 9. ABC is a triangle, and through D, the middle point of AB, DE, DF are drawn parallel to the sides BC, AC, to meet them in E, F. Shew that EF is parallel to AB. 10. Through a given point to draw a line such that the part of it intercepted between two parallel lines shall have a given length. 11. To describe a rhombus equal to a given parallelogram, having its side equal to the longer side of the parallelogram. 12. Shew that the diagonal of a rectangle is longer than any other line whose extremities are on the sides of the rectangle. 13. From the extremities of the base of an isosceles triangle straight lines are drawn perpendicular to the opposite sides; the angles made by them with the base are equal to half the vertical angle. 14. D is the middle point of the side AC of a triangle ACB, and any parallel lines BE, DF are drawn to meet AC, AB (or BC) in E and F, shew that EF divides the triangle into two equal areas. 15. If every pair of alternate sides of a convex figure of five sides be produced to meet, so as to form a five-rayed star, prove that the angles so formed will be together equal to two right angles. Extend this to the case of a polygon of n sides. 16. Of all triangles having the same base and area, that which is isosceles has the least perimeter. 17. The area of a rhombus is equal to half the rectangle constructed on the two diameters of the rhombus. 18. If two opposite sides of a quadrilateral are parallel, and their points of bisection joined, the quadrilateral will be bisected. 19. If two opposite sides of a parallelogram be bisected, and lines be drawn from these two points of bisection to the opposite angles, these lines will be parallel, two and two, and will trisect both diagonals. .. 20. The sum of the squares described on the sides of a rhombus is equal to the squares described on its diameters. 21. From the sides of the triangle ABC, AA', BB', CC', are cut off each equal to two-thirds of the side from which it is cut. Shew that the triangle A'B'C' is one-third of the triangle ABC. 22. BCD... are points on the circumference of a circle, A any point not the centre of the circle. Shew that of the lines AB, AC, AD... not more than two can be equal. . 23. Find the locus of a point, such that the sum of the squares on its distances from two given points is equal to the square on the distance between the two points. 24. If m and n are any numbers, and lines be taken whose lengths are ma + no, ma – na and 2mn units respectively, shew that these lines will form a right-angled triangle. Give examples of these triangles. 25. Through two given points on opposite sides of a straight line draw two straight lines to meet in that line, so that the angle which they form shall be bisected by that line. 26. Through a given point draw a line such that the perpendiculars on it from two given points may be equal. 27. Find points D, E in the equal sides AB, AC of an isosceles triangle ABC, such that BD=DE= EC. 28. If one angle of a triangle is equal to the sum of the other two, the greatest side is double of the distance of its middle point from the opposite angle. 29. Find the locus of a point, given the sum or difference of its distances from two fixed lines. 30. Given two points and a straight line of indefinite length, construct an equilateral triangle so that two of its sides shall pass through the given points, and the third shall be in the given straight line. 31. Construct an isosceles triangle having the angle at the vertex double of the angles at the base. 32. ABC is a triangle, AB greater than BC; BD bisects the base AC, and BE the angle ABC. Prove (1) that ADB is an obtuse angle; (2) that ABD is less than DBC; and (3) that BE is less than BD. |