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Euclid Papers set in the Mathematical Tripos at Cambridge from 1848 to 1872.
QUESTIONS arising out of the Propositions, to which they are attached, have been proposed in the Euclid Papers to Candidates for Mathematical Honours since the year 1848.
A complete set of these questions, so far as they refer to Books I.-IV., is here given. The figures preceding each question denote the particular Proposition to which the question was attached. It is expected that the solution of each question is to be obtained mainly by using the Proposition which precedes it, and that no Proposition which comes later in Euclid's order should be assumed.
Of some of the questions here given we have already made use in the preceding pages. As examples, however, of what has been hitherto expected of Candidates for Honours, and in order to keep the series of Papers complete, we have not hesitated to repeat them.
1848. I. c. How does it appear that the two triangles are equiangular and equal to each other?
1. 34. If the two diagonals be drawn, shew that a
parallelogram will be divided into four equal
parts. In what case will the diagonal bisect the angle of parallelogram?
III. 15. Shew that all equal straight lines in a circle will be touched by another circle.
III. 20. If two straight lines AEB, CED in a circle intersect in E, the angles subtended by AC and BD at the centre are together double of the angle AEC.
1. 1. By a method similar to that used in this problem, describe on a given finite straight line an isosceles triangle, the sides of which shall be each equal to twice the base.
п. 11. Shew that in Euclid's figure four other lines beside the given line, are divided in the required manner.
IV. 4. Describe a circle touching one side of a triangle and the produced parts of the other two.
I. 34. If the opposite sides, or the opposite angles, of any quadrilateral figure be equal, or if its diagonals bisect each other, the quadrilateral is a parallelogram.
II. 14. Given a square, and one side of a rectangle which is equal to the square, find the other
III. 31. The greatest rectangle that can be inscribed in a circle is a square.
III. 34. Divide a circle into two segments such that the angle in one of them shall be five times the angle in the other.
IV. 10. Shew that the base of the triangle is equal to the side of a regular pentagon inscribed in the smaller circle of the figure.
1. 38. Let ABC, ABD be two equal triangles, upon the same base AB and on opposite sides of
it: join CD, meeting AB in E: shew that CE is equal to ED.
J. 47. If ABC be a triangle, whose angle A is a right angle, and BE, CF be drawn bisecting the opposite sides respectively, shew that four times the sum of the squares on BE and CF is equal to five times the square on BC.
III. 22. If a polygon of an even number of sides be inscribed in a circle, the sum of the alternate angles together with two right angles is equal to as many right angles as the figure has sides.
1851. Iv. 16. In a given circle inscribe a triangle, whose angles are as the numbers 2, 5 and 8.
L. 42. Divide a triangle by two straight lines into three parts, which, when properly arranged, shall form a parallelogram whose angles are of given magnitude.
11. 12. Triangles are described on the same base and having the difference of the squares on the
other sides constant: shew that the vertex of any triangle is in one or other of two fixed straight lines.
IV. 3. Two equilateral triangles are described about the same circle: shew that their intersections
will form a hexagon equilateral, but not generally equiangular.
1853. 1.B. Cor. If lines be drawn through the extremities of the base of an isosceles triangle, making angles
with it, on the side remote from the vertex, each equal to one third of one of the equal angles, and meeting the sides produced, prove that three of the triangles thus formed are isosceles.
1. 29. Through two given points draw two lines, forming with a line, given in position, an equilateral triangle.
II. 11. In the figure, if H be the point of division of the given line AB, and DA be the side of the square which is bisected in E and produced to F, and if DH be produced to meet BF in L, prove that DL is perpendicular to BF, and is divided by BE similarly to the given line. III. 32. Through a given point without a circle draw a chord such that the difference of the angles
in the two segments, into which it divides the circle, may be equal to a given angle.
III. 36. From a given point as centre describe a circle cutting a given line in two points, so that the rectangle contained by their distances from a fixed point in the line may be equal to a given square
1. 43. If K be the common angular point of the parallelograms about the diameter, and BD the other diameter, the difference of the parallelograms is equal to twice the triangle BKD. II. 11. Produce a given straight line to a point such that the rectangle contained by the whole line thus produced and the part produced shall be equal to the square on the given straight line.
III. 22. If the opposite sides of the quadrilateral be produced to meet in P, Q, and about the triangles so formed without the quadrilateral circles be described meeting again in R, shew that P, R, Q will be in one straight line.
iv. 10. Upon a given straight line, as base, describe an isosceles triangle having the third angle
treble of each of the angles at the base.
1855. 1. 20. Prove that the sum of the distances of any point from the three angles of a triangle is greater than half the perimeter of the triangle.
I. 47. If a line be drawn parallel to the hypotenuse
double of the square on the other, without
III. 27. If any number of triangles, upon the same base
BC, and on the same side of it, have their vertical angles equal, and perpendiculars meeting in D be drawn from B, C upon the opposite sides, find the locus of D, and shew that all the lines which bisect the angle BDC pass through the same point.
1855. IV. 4. If the circle inscribed in a triangle ABC touch the sides AB, AC in the points D, E, and a straight line be drawn from A to the centre of the circle, meeting the circumference in G, shew that G is the centre of the circle inscribed in the triangle ADE.
1856. 1. 34. O all parallelograms, which can be formed with diameters of given length, the rhombus is
II. 12. If AB, one of the equal sides of an isosceles triangle ABC, be prod iced beyond the base to D, so that BD=AB, shew that the square on CD is equal to the square on AB together with twice the square on BC.
IV. 15. Shew how to derive the hexagon from an equilateral triangle inscribed in the circle, and from this construction shew that the side of the hexagon equals the radius of the circle, and that the hexagon is double of the triangle.
1857. 1. 35. ABC is an isosceles triangle, of which A is the vertex: AB, AC are bisected in D and E respectively; BE, CD intersect in F: shew that the triangle ADE is equal to three times the triangle DEF.
II. 13. The base of a triangle is given, and is bisected by the centre of a given circle, the circum
ference of which is the locus of the vertex :
prove that the sum of the squares on the two sides of the triangle is invariable.
III. 22. Prove that the sum of the angles in the four segments of the circle, exterior to the quadrilateral, is equal to six right angles.
IV. 4. Circles are inscribed in the two triangles formed by drawing a perpendicular from an angle of a triangle upon the opposite side, and analogous circles are described in relation to the two other like perpendiculars prove that the