PROPOSITION XVI. PROBLEM.

To inscribe a regular quindecagon in a given circle.

E

B

Let ABC be the given O.

It is required to inscribe in the a regular quindecagon. Let AB be the side of an equilateral ▲ inscribed in the O,

IV. 2.

and AD the side of a regular pentagon inscribed in the .

IV. 11.

Then of such equal parts as the whole Oce ABC contains fifteen,

arc ADB must contain five,

and arc AD must contain three,

and.. arc DB, their difference, must contain two.

Bisect arc DB in E.

III. 30.

Then arcs DE, EB are each the fifteenth part of the whole

Oce.

If then chords DE, EB be drawn,

and chords equal to them be placed all round the Oce, IV. 1. a regular quindecagon will be inscribed in the O.

Q. E. F.

Miscellaneous Exercises on Book IV.

1. The perpendiculars let fall on the sides of an equilateral triangle from the centre of the circle, described about the iangle, are equal.

2. Inscribe a circle in a given regular octagon.

3. Shew that in the diagram of Prop. X. there is a second triangle, which has each of two of its angles double of the third. 4. Describe a circle about a given rectangle.

5. Shew that the diameter of the circle which is described about an isosceles triangle, which has its vertical angle double of either of the angles at the base, is equal to the base of the triangle.

6. The side of the equilateral triangle, described about a circle, is double of the side of the equilateral triangle, inscribed in the circle.

7. A quadrilateral figure may have a circle described about it, if the rectangles contained by the segments of the diagonals be equal.

8. The square on the side of an equilateral triangle, inscribed in a circle, is triple of the square on the side of the regular hexagon, inscribed in the same circle.

9. Inscribe a circle in a given rhombus.

10. ABC is an equilateral triangle inscribed in a circle; tangents to the circle at A and B meet in M. Shew that a diameter drawn from M bisects the angle AMB, and is itself trisected by the circumference.

11. Compare the areas of two regular hexagons, one inscribed in, the other described about, a given circle.

12. Inscribe a square in a given semicircle.

13. A circle being given, describe six other circles, each of them equal to it, and in contact with each other and with the given circle.

14. Given the angles of a triangle, and the perpendiculars from any point on the three sides, construct the triangle.

15. Having given the radius of a circle, determine its centre, when the circle touches two given lines, which are not parallel.

16. If the distance between the centres of two circles, which cut one another at right angles, is equal to twice one of the radii, the common chord is the side of the regular hexagon, inscribed in one of the circles, and the side of the equilateral triangle, inscribed in the other.

17. Construct a square, having given the sum, or the difference, of the diagonal and the side.

18. If from O, the centre of the circle inscribed in a triangle ABC, OD, OE, OF be drawn perpendicular to the sides BC, CA, AB, respectively, and from any point P in OP, drawn parallel to AB, perpendiculars PQ, PR be drawn upon OD and OE respectively, or these produced, shew that the triangle QRO is equiangular to the triangle ABC.

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?

I. 34. If the two diagonals be drawn, shew that a

paral'clogram 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.

1849. 1. 1. By a method similar to that used in this problem, describe on a given finite straight line

1850.

1851.

an isosceles triangle, the sides of which shall be each equal to twice the base.

II. 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.

1. 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

side.

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.