2.

On a given straight line describe a regular hexagon.

3. Find a point which is equally distant from three given points. Is this problem always possible?

4. Inscribe a circle in a rhombus.

Can a circle be described about a rhombus ?

5. Describe a parallelogram about a circle, having an angle equal to a given angle.

6. If a parallelogram be described about a circle, it will be equilateral.

7. Determine the condition that the straight line joining the centres of the circles inscribed in, and described about a given triangle, may pass through one of the angular points of the triangle.

8.

What relation exists between the sides of a triangle when the centres of the circles inscribed in and described about the triangle coincide?

9.

Describe a regular hexagon having its angular points in the sides of a given equilateral triangle.

IO. If the chords which bisect two angles of a triangle inscribed in a circle be equal, prove that either the angles are equal or else the third angle is equal to the angle of an equilateral triangle.

II. If one square be inscribed in another, the dif ference between their areas is equal to twice the rectangle contained by the segments of any one of the sides of the larger square.

I2. If one square be inscribed in another, the area of the smaller square is equal to the squares on the segments of a side of the larger square.

13. Apply the previous deduction to shew how the square described on the hypothenuse of a right-angled triangle may be divided into parts which, when properly arranged, will form the squares on the sides containing the right angle.

14. If a quadrilateral figure be circumscribed about a circle, then two of its opposite sides are together equal to the other two.

15. If two of the figure are equal to the inscribed within it.

opposite sides of a quadrilateral other two, then a circle can be

16. If a quadrilateral figure be circumscribed about a circle, then will the angles subtended at the centre by two of its opposite sides be together equal to two right angles. Three circles touch each other externally, prove 17. that their centres are the angular points of the triangle described about the circle which passes through the points of contact.

18. The circles each of which touches two sides of a regular pentagon at the extremity of a third meet in a point.

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.

2I. 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.

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.

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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 : 2 : 3: 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.

IO. 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 triangle 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