PROBLEMS. BOOKS V. XI. XII. AND MISCELLANEOUS. 1. IF any three lines be drawn making equal angles with the sides of a triangle toward the same parts, they will form a triangle similar to the former. 2. The vertical angle BAC of a triangle is bisected by a line on which the perpendiculars BD, CE are drawn bisect the base BC in F, and shew that FD=FE. 3. From the vertex of an isosceles triangle, two lines are drawn to the opposite angles of the square upon the base: shew that the line, which joins their intersections with the diagonals, is parallel to the base. 4. In the figure 1. 47, shew that the lines BC, FK, GH, meet in a point. 5. If England be a triangle, whose base is 200 miles and altitude 300 miles, and if the population be 20 millions, how many persons are there to an acre? 6. The sum of the perpendiculars, let fall from any point within an equilateral triangle upon the sides, will be equal to that let fall from one of its angles upon the opposite side. Is this true when the point is in one side of the triangle? How must the statement be made when the point is without the triangle? 7. If P be any point in the plane of a parallelogram, ABCD, shew that the triangle PBD is equal to the sum of the triangles PAB, PBC. 8. Let a, b, c represent the sides opposite to the angles A, B, C, of any triangle; then if Ã be a right angle, and the length of the perpendicular from A on BC, the triangle whose sides are b+c, h, a+h, is also rightangled. 9. Express AD, the perpendicular upon the base BC of any triangle, in terms of a, b, c; and shew that if S represent the semisum of the sides, the area = √{S (S-a) (S−b) (S−c)}. 10. If an equilateral triangle have its angular points in three parallel lines, of which the middle one is distant from the outside ones by b, c, its side will be 11. If the sides of a pentagon be produced both ways to meet, shew that the sum of the angles at the points of intersection is equal to two right angles; and, generally, if a polygon have n+4 sides, shew that the sum of the angles thus formed is equal to 2n right angles. 12. Shew that the triangle, square, and hexagon, are the only regular polygons, by which space can be completely filled up about a point. 13. Shew how the squares in (1. 47) may be dissected, so that its truth may be made to appear by superposition of the parts. 14. If AD be drawn to any point D in the base of the triangle ABC, shew that AB2.CD+AC2.BD=AD2.BC+BC.BD.CD. 15. Find the locus of a point, when (i) the sum, and (ii) the difference, of its squares from two given points is constant. 16. C is any point in AB; find a point D (i) in CB, (ii) in CB produced, such that CD may be a mean proportional between AD, BD. 17. AB is divided in extreme and mean ratio at C: in BC (the greater segment) produced take BD=BC, and in BC take BE-AC; then will AD, AE be also similarly divided in B, C. 18. ABCD is a trapezium, AD parallel to BC: shew that AC2-BD2: AB2~ CD2 :: BC+AD: BC~AD. 19. Divide algebraically a given line a into two parts, such that the rectangle of the whole and one part may be equal to the square of the other part. Deduce Euclid's construction from one solution, and explain the other. 20. Two circles, whose radii are as 2:3, touch each other internally, and through the centre of the smaller circle a line is drawn perpendicular to the common diameter: shew that tangents drawn to this circle, from the points where the line cuts the larger circle, are perpendicular to each other. 21. In any inscribed polygon of an even number of sides, the sum of the 1st, 3rd, 5th, &c. angles is equal to the sum of the 2nd, 4th, 6th, &c. angles. 22. BD, CD, are drawn perpendicular to the sides AB, AC of a triangle, and CFE, perpendicular to AD, cuts AB in E: shew that the triangles ABC, ACE are similar. 23. AB, the line of centres of two circles, whose radii are R, r, is divided in C, so that AB: R+r:: R-r: AC-BC: shew that the tangents drawn to the circle from any point in CD, perpendicular to AB, are equal. 24. Describe a rectangle equal to a given square, and having the difference of its adjacent sides equal to a given line. 25. If a circle be described about a triangle ABC, and perpendiculars from the angles on the sides cut the circle in D, E, F, the arcs DE, ŎF, EF, will be bisected in C, B, A. 26. If any two circles touch in O, and lines, POP', QOQ, be drawn at right angles to each other, then, A, A', being the points in which the line of centres cuts the circles, PP'2+QQ"2=AA'2. A A 27. AOB is a quadrant: draw any chord QRrg parallel to AB, cutting OA, OB in R, r, and shew that QR2 + Qr2 = AB2. 28. Find a point such that tangents drawn from it to touch two given circles shall contain a given angle. 29. If from any point in the circumference of a circle perpendiculars be drawn upon the sides of the inscribed triangle, the three points of intersection will be in the same line. 30. AB, AC, DE, DF, are four lines, which form by their intersections, three and three together, four triangles: shew that the circles which circumscribe them will pass through one and the same point. 31. If the base of a triangle be produced both ways, so that each part produced may equal the adjacent side, and through the extremities of the parts produced and vertex a circle be described, the line joining the vertex and centre of the circle will bisect the vertical angle. 32. No triangle can be inscribed in a square greater than half the square. 33. If R, r be the radii of the circumscribed and inscribed circles of a triangle, shew that its area 34. The sides of a triangle are in Ar. Prog, a, α being the longest and shortest sides: if R, r be the radii of the circumscribed and inscribed circles, shew that 6 Rraa'. 35. ABCD is an inscribed quadrilateral: shew that AC: BD:: BA.AD+BC.CD: AB.BC+AD.DC. 36. If the circumference of a circle be divided into five equal parts in the points A, B, C, D, E, and these be joined two and two, that is, AC, CE, EB, BD, DA, shew that the angles of the stellated figure thus formed are together equal to two right angles. 37. If the number of parts in [36] be m, and the points of division be joined n and n together, n being less than 1m, shew that the of the stellated figure thus formed are together equal to 2 (a2b) right angles, where a, b are the prime factors of m and n. How may this result be applied to the case when ʼn is greater than m? 38. Construct the figures for stellated pentagons, hexagons, heptagons, and octagons; and determine how many forms are possible in each case. 39. If AB be divided in extreme and mean ratio at C, shew that the ratio of AC: CB is incommensurable, but nearly 13. 40. A ratio of greater inequality is diminished, and of less inequality increased, by adding the same quantity to both its terms. 41. Apply a property of the circle (III. 15) to shew that, if four lines are proportionals, the sum of the greatest and least will be greater than the sum of the other two. 42. If, from any point O within a triangle, there be drawn Oa, Ob, Oc, to the sides, and from the angles Aa', Bb', Ce', be drawn parallel to these, shew that Oa об Oc + + =1. Aa' Bo' Cc' 43. Construct a right-angled triangle, having given the sum of the base and perpendicular, and the sum of the base and hypothenuse. 44. If A be the area of any triangle, shew that the area of a triangle, whose angular points divide the sides n2-n+1 of the former in the ratio of n. 1, will be (n+1)2 .A. 45. From the right angle A of a triangle a perpendicular Ap is dropped upon BC, pq on AB, qr on BC, &c.: shew that Ap+pq+qr+&c. : AB :: AB+BC: AC. |