1 APPENDIX-(A.) EXERCISES ON EUCLID, BOOKS I. TO IV., CAREFULLY SELECTED FROM THE Cambridge Mathematical Tripos Fan. 1867. 1. If one side of a triangle be produced, the exterior angle is greater than either of the two interior opposite angles. Any two exterior angles of a triangle are together greater than two right angles. 2. The complements of parallelograms, which are about the diameter of any parallelogram, are equal to each other. What is the greatest value which these complements, for a given parallelogram, can have? 3. Divide a given straight line into two parts, such that the rectangle contained by the whole and one of the parts shall be equal to the square on the other part. Divide a given straight line into two parts, such that the squares on the whole line and on one of the parts shall be together double of the square on the other part. 4. The opposite angles of a quadrilateral inscribed in a circle are together equal to two right angles. If the chords, which bisect two angles of a triangle inscribed in a circle, be equal, prove that either the angles are equal, or the third angle is equal to the angle of an equilateral triangle. Fan. 1868. 1. If a parallelogram and a triangle be on the same base and between the same parallels, the parallelogram shall be double of the triangle. OKBM and OLDN are parallelograms about the diameter of a parallelogram ABCD. In MN, which is parallel to BA, take any point P and prove that, if PC, produced if necessary, meet KL in Q, BP will be parallel to DQ. 2. In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle, by twice the rectangle contained by the side on which, when produced, the perpendicular falls, and the straight line intercepted without the triangle, between the perpendicular and the obtuse angle. In a triangle ABC, D, E, F are the middle points of the sides BC, CA, AB respectively, and K, L, M are the feet of the perpendiculars on the same sides from the opposite angles. Prove that the greatest of the rectangles contained by BC and DK, CA and EL, AB and FM is equal to the sum of the other two. 3. If two straight lines cut one another within a circle, the rectangle contained by the segments of one of them shall be equal to the rectangle contained by the segments of the other. Through a point within a circle draw a chord, such that the rectangle contained by the whole chord and one part may be equal to a given square. Determine the necessary limits to the magnitude of this square. 4. Inscribe a circle in a given triangle. If two triangles ABC, A1BC1 be inscribed in the same circle, so that AД1, BB1, CC1 meet in one point 0; prove that, if O be the centre of the inscribed circle of one of the triangles, it will be the centre of perpendiculars of the other. A circle A passes through the centre of a circle B, prove that their common tangents will touch A in points lying on a tangent to B. In a given quadrilateral a parallelogram is inscribed, whose sides are parallel to the diagonals of the quadrilateral; prove that the diagonals of all such parallelograms intersect on the line which joins the middle points of the diagonals of the quadrilateral, and that the area of the greatest of such parallelograms is half that of the quadrilateral. Fan. 1869. 1. Equal triangles on equal bases in the same straight line and on the same side of it, are between the same parallels. ABC is a triangle, E and F two points; if the sum of the triangles ABE and BCE be equal to the sum of the triangles ABF and BCF, then under certain conditions EF will be parallel to AC. Find these conditions, and determine when the difference instead of the sum of the triangles must be taken. 2. Divide a line into two parts, so that the rectangle contained by the whole and one of the parts shall be equal to the square of the other part. Show that the point of section lies between the extremities of the line. 3. Upon a given straight line describe a segment of a circle which shall contain an angle equal to a given rectilineal angle. An acute-angled triangle is inscribed in a circle, and the paper is folded along each of the sides of the triangle: show that the circumferences of the three segments will pass through the same point. State the equivalent proposition for an obtuseangled triangle. 4. Inscribe an equilateral and equiangular pentagon in a given circle. Show that the circles, each of which touches two sides at the extremities of a third, meet in a point. Given the inscribed and circumscribed circles of a triangle, the centre of perpendiculars will lie on a fixed circle. If from any point, straight lines be drawn to the angles of a triangle, and if through each of these angles straight lines be drawn, making with either of the sides an angle equal to that which the line drawn from the point makes with the other side, these three straight lines will meet in a point. On two straight lines not in the same plane are taken points A, B, C; A1, B1, C1 respectively, prove that the three straight lines, each of which bisects both straight lines of one of the pairs of BC1, B1C; CA1, CA; AB, A1B will meet in a point. Two chords AB, AC of a circle are drawn; the perpendicular from the centre on AB meets AC in D: prove that the straight line joining D to the pole of BC is parallel to AB. Given the centres of four circles, determine their radii, so that they may pass, three and three, through four points. Fan. 1870. 1. If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, namely, either the sides adjacent to the equal angles, or sides which are opposite to equal angles in each, then shall the other sides be equal, each to each, also the third angle of the one equal to the third angle of the other. ABCD is a square, and E a point in BC, a straight line EF is drawn at right angles to AE, and meets the straight line, which bisects the angle between CD and BC produced, in a point F; prove that AE is equal to EF. 2. If a straight line be divided into two equal, and also into two unequal parts, the squares on the two unequal parts are together double of the square on half the line and of the square on the line between the points of section. The diagonals of a quadrilateral meet in E, and F is the middle point of the straight line joining the middle points of the diagonals: prove that the sum of the squares on the straight lines joining E to the angular points of the quadri H |