diameter describe a semicircle, U being the centre, and produce RQ to meet the circumference in V.

If a square be described on QV, this square will be equal to ABCDE.

Proof. For since PQ is the sum of PU and UQ, and QT is the difference of PU (or UT) and UQ, it follows (from Theorem 27) that the rectangle PQx QT-PU' — UQ'; but PU2 = UV2, and therefore PU1– UQ3=UV3 – UQ2, that is, VQ, by Theorem 30.

=

But the rectangle PQ× QT is the rectangle PQRS, which was made equal to ABCDE.

=

Therefore VQ ABCDE, and the square described on VQ is the square required.

Remark. If the given figure is not rectilineal, it cannot be divided into triangles: hence by this method it is impossible to construct a square equal to a given curvilinear area. Nor can any method depending on the use of the ruler and compasses only, (see p. 38,) construct a square equal to some curvilinear areas, such as the circle. This is the problem of squaring the circle, the solution of which cannot be effected without the use of other instruments.

We subjoin a few problems and theorems as miscellaneous exercises in the Geometry of angles, lines, triangles, parallelograms, and the equivalence of figures.

I.

MISCELLANEOUS THEOREMS AND PROBLEMS.

Prove that the acute angle between the bisectors of the angles at the base of an isosceles triangle is equal to one of the angles at the base of the triangle.

2. Find a point equally distant from three given straight lines.

3. Find the locus of the middle point of a line drawn from a given point to meet a given line.

4. If the diagonals of a quadrilateral bisect one another and are equal to one another, the figure will be a rectangle.

5. If the diagonals of a quadrilateral bisect one another at right angles and are also equal, the figure will be a square.

6. If ABC is a triangle, AB being greater than AC, and a point D in AB be taken such that AD = AC; prove that BCD is equal to half the difference of the angles ABC, ACB.

7. If the opposite angles of a quadrilateral are equal, the figure is a parallelogram.

8. If ABCD is a parallelogram, and AE= CF are cut off from the diagonal AC, then BEDF will be a parallelogram.

=

9. If AA CC be cut off from the diagonal AC, and BB = DD from the diagonal BD of a parallelogram, then will A'B'C'D' be also a parallelogram.

IO. If AA=BB = CC' = DD' be cut off from the sides of the parallelogram ABCD taken in order, then will A'B'C'D' be also a parallelogram.

II. ABC is a triangle, and through D, the middle point of AB, DE, DF are drawn parallel to the sides BC, AC to meet them in EF. Shew that EF is parallel to AB.

12. Through a given point to draw a line such that the part of it intercepted between two parallel lines shall have a given length.

13. To describe a rhombus equal to a given parallelogram, having its side equal to the longer side of the parallelogram.

14. Shew that the diagonal of a rectangle is longer than any other line whose extremities are on the sides of the rectangle.

15. From the extremities of the base of an isosceles triangle straight lines are drawn perpendicular to the opposite sides; the angles made by them with the base are equal to half the vertical angle.

16. If one angle of a triangle is equal to the sum of the other two, the greatest side is double of the distance of its middle point from the opposite angle.

17. Find the locus of a point, given the sum or difference of its distances from two fixed lines.

18. ABC is a triangle, AB greater than BC; BD bisects the base AC, and BE the angle ABC. Prove (1) that ADB is an obtuse angle; (2) that ABD is less than DBC; and (3) that BE is less than BD.

19. If two sides of a triangle be given, its area will be greatest when they contain a right angle.

20. BCD... are points on the circumference of a circle, A any point not the centre of the circle. Shew that of the lines AB, AC, AD... not more than two can be equal.

21. Of all triangles having the same base and area, that which is isosceles has the least perimeter.

22. Of all triangles having the same vertical angle, and whose bases pass through a given point, the least is that whose base is bisected in that point.

23. The diagonals of a parallelogram divide it into four equivalent triangles.

24. If from any point in the diagonal of a parallelogram straight lines be drawn to the angles, then the parallelogram will be divided into two pairs of equivalent triangles.

25. ABCD is a parallelogram, and E any point in the diagonal AC produced. Shew that the triangles EBC, EDC will be equivalent.

26. ABCD is a parallelogram, and O any point within it, shew that the triangles OAB, OCD are together equivalent to half the parallelogram.

27. On the same supposition if lines are drawn through O parallel to the sides of the parallelogram, then the difference of the parallelograms DO, BO is double of the triangle OAC.

28. The diagonals of a parallelogram intersect in O, and P is a point within the triangle OAB. Prove that the difference of the triangles APB, CPD, is equivalent to the sum of the triangles APC, BPD.

29. If the points of bisection of the sides of a triangle be joined, the triangle so formed shall be one-fourth of the given triangle.

30.

Shew that the sum of the squares on the lines

joining the angular points of a square to any point within it is double of the sum of the squares on the perpendiculars from that point on the sides.

31. If the sides of a quadrilateral figure be bisected, and the points of bisection joined, prove that the figure so formed will be a parallelogram equal in area to half the given quadrilateral.

32. Any line drawn through the intersection of the diagonals of a parallelogram to meet the sides bisects the figure.

33. D is the middle point of the side AC of a triangle ACB, and any parallel lines BE, DF are drawn to meet AC, AB (or BC) in E and F, shew that EF divides the ria ngle into two equal areas.

34. If the sides of a triangle are 3, 4, 5 inches respectively, the triangle is right-angled.

35. The area of a rhombus is equal to half the rectangle constructed on the two diameters of the rhombus.

36. If two opposite sides of a quadrilateral are parallel, and their points of bisection joined, the quadrilateral will be bisected.

37. If two opposite sides of a parallelogram be bisected, and lines be drawn from these two points of bisection to the opposite angles, these lines will be parallel, and will trisect the diagonal.

38. The sum of the squares described on the sides of a rhombus is equal to the squares described on its diameters.

39. From the sides of the triangle ABC, AA', BB', CC', are cut off each equal to two-thirds of the side from which it is cut. Shew that the triangle A'B'C' is one-third of the triangle ABC.