19. If two diagonals of a regular pentagon be drawn cutting one another, the greater segments will each be equal to a side of the pentagon.

20. An equilateral figure inscribed in a circle is also equiangular.

21. An equiangular figure is inscribed in a circle: prove that the alternate sides are equal.

22.

An equiangular figure having an odd number of sides is inscribed in a circle: prove that it is equilateral.

23. On the base of a triangle describe an isosceles triangle whose vertical angle shall be double that of the given triangle. Is this problem always possible?

24.

Describe a circle passing through one of the angular points of a rhombus, and touching the sides containing the opposite angle.

BOOK VI.

I.

1. Through a given point draw a straight line which shall be divided in a given ratio by two given straight lines.

2. If perpendiculars be let fall from the extremities of one straight line upon another, prove that their feet are equidistant from the middle point of the former straight line.

3. Divide a straight line into two parts such that one of them shall be four times as long as the other.

4. Given the perimeter and the vertical angle of an isosceles triangle, construct it.

C. G.

17

5. From two straight lines cut off two parts having a given ratio, so that the sum of the squares of the remainders may be equal to a given square.

6. If from any point in the circumference of the exterior of two concentric circles, two straight lines be drawn touching the interior and cutting the exterior, the distance between the points of contact will be half that between the points of intersection.

7. PQRS is a square, PR its diagonal; bisect PS in T, and join QT cutting PR in K, then RK = twice PK; and ▲ PTK : ^ RTK:▲ PQT: ≤ QRK as 1 : 23: 4.

8. ABC, ADE are two isosceles triangles having a common angle at A. Through B, E draw BG, EF | to one another; join DG, CF. Then shall DG be to CF.

9. If RPS, RQS are two triangles on RS such that RP: PS as RQ: QS,

then the lines bisecting the angles at P, Q intersect RS in the same point.

10. A point is taken in one of the sides of a triangle, and through it a line is drawn to another side meeting the third side, and through the point of intersection another line to the first side, and so on; shew that at the end of the second revolution the line will pass through the original point.

II. If points D, E be taken in the sides BC, CA of a triangle, such that BD: DC as BA : AE, and DG be drawn to CA meeting BE in G; prove that AG will . bisect the angle BAC.

12. D is a point in the side AB of the obtuse-angled angle ABC having the obtuse angle at A, such that

CD is a fourth proportional to the sides AB, BC, CA; prove that the triangles ABC, ADC are similar.

13. Describe a circle which shall touch two given straight lines and pass through a given point.

14. To inscribe a square in a given sector of a circle.

15.

Given the difference between the diagonal and side of a square; construct the square.

16. Find a point D in the base BC of a triangle ABC such that AD may be a mean proportional between AB and AC.

17. Describe a rhombus equal and equiangular to a given parallelogram.

18. Of all equal and equiangular parallelograms shew that the rhombus has the least perimeter.

19. Produce a straight line which is divided into two parts to a point such that the whole line shall be divided harmonically.

20. Divide a given straight line harmonically.

21. Of all equal rhombuses the square has the least perimeter.

22. Of all equal quadrilateral figures the square has the least perimeter.

23. If two circles touch each other, and also touch a given straight line, the part of the tangent between the points of contact is a mean proportional between the diameters of the circles.

24. In any right-angled triangle one side is to the other as the excess of the hypothenuse above the second

is to the line cut off from the first between the right angle and the line bisecting the opposite angle.

25. Describe a square which shall bear to another a given ratio.

26. In a regular pentagon inscribe a square.

27. Upon a given base describe a triangle having a given vertical angle and one of its sides double of the other.

28. Inscribe in a segment of a circle a rectangle one of whose sides is treble each of the adjacent sides.

29. Divide a triangle into four equal and similar triangles.

30. Describe a triangle similar to a given triangle and having its perimeter equal to a given straight line.

31. The semicircle described on the hypothenuse of a right-angled triangle is equal to the sum of the semicircles described upon the sides.

32. If a straight line be divided into any two parts, and similar figures be described on the whole line and on the parts, then will the perimeter of the former figure be equal to the sum of the perimeters of the latter figures.

33. Two similar rectilineal figures being given, describe another whose perimeter shall be equal to the sum of the perimeters of the two given figures.

34. Two similar rectilineal figures being given, describe another whose perimeter shall be equal to the difference of the perimeters of the two given figures.

35. ABCD is a quadrilateral inscribed in a circle; its diagonals intersect in X: prove that

▲ ABX: ▲ DCX as sq. on BX: sq. on CX.

36. If two diagonals of a pentagon be drawn to cut one another, they will each be divided in extreme and mean ratio.

37. APB is a quadrant, of which C is the centre, SPT a straight line touching the arc at P meeting CB, CA produced in S, T, and PM is perpendicular to CA; prove that

A SCT: ACB as ▲ ACB: ▲ CMP.

38. From a given point in the side of a triangle to draw straight lines which shall divide the triangle into a given number of equal parts.

39. If lines be drawn from a given point in the circumference of a circle, such that the rectangle between the whole lines and the parts of them within the circle is constant, the locus of their extremities will be a straight line.

40.

Given the base, the ratio of the sides containing the vertical angle, and the distance of the vertex from a given point in the base, to construct the triangle.

41. Two similar figures being given, describe a similar figure equal to their sum.

42. Two similar figures being given, describe a similar figure equal to their difference.

43. Describe a figure similar to a given figure, and bearing to it a given ratio.