Next, let the sides which are opposite to equal angles in each triangle be equal, viz. AB=DE.

Then_must_AC=DF, and BC=EF, and L BAC= 1 EDF.

For if BC be not=EF, let BC be the greater, and make BH=EF, and join AH.

Then in As ABH, DEF,

:: AB=DE, and BH=EF, and ▲ ABH= ▲ DEF,

:. LAHB= LDFE.

But ACB=LDFE, by hypothesis,

:. LAHB= L ACB;

I. 4.

that is, the exterior of ▲ AHC is equal to the interior and opposite ACB, which is impossible.

:. BC is not greater than EF.

In the same way it may be shewn that BC is not less than EF;

:. BC=EF.

Then in As ABC, DEF,

:: AB=DE, and BC= EF, and ▲ ABC= ▲ DEF,

:. AC=DF, and BAC=1 EDF.

Q. E. D.

Miscellaneous Examples on Books I. and II.

1. AB and CD are equal straight lines bisecting one another at right angles. Shew that ACBD is a square.

2. From a point in the side of a parallelogram draw a line dividing the parallelogram into two equal parts.

3. Draw through a point lying between two lines that intersect a line terminated by the given lines and bisected in the given point.

4. The square on the hypotenuse of an isosceles rightangled triangle is equal to four times the square on the perpendicular from the right angle on the hypotenuse.

5. Describe a rhombus which shall be equal to a given triangle and have each of its sides equal to one side of the triangle.

6. Shew how to describe a square when the difference between the lengths of a diagonal and a side is given.

7. Two rings slide on two straight lines which intersect at right angles in a point O, and are connected by an inextensible string passing round a peg fixed at that point. Shew that the rings will be nearest to each other when they are equidistant from 0.

8. ABCD is a parallelogram, whose diagonals AC, BD intersect in O; shew that if the parallelograms AOBP, DOCQ be completed, the straight line joining P and Q passes through O.

9. ABCD, EBCF are two parallelograms on the same base BC and so situated that CF passes through A. Join DF, and produce it to meet BE produced in K; join FB, and prove that the triangle FAB equals the triangle FEK.

10. The alternate sides of a polygon are produced to meet; shew that all the angles at their points of intersection together with four right angles are equal to all the interior angles of the polygon.

11. Shew that the perimeter of a rectangle is always greater than that of the square equal to it.

12. Shew that the opposite sides of an equiangular hexagon are parallel, though they be not equal; and that any two sides that are adjacent are together equal to the two which are parallel.

13. If two equal straight lines intersect each other anywhere at right angles, shew that the area of the quadrilateral formed by joining their extremities is invariable and equal to one-half the square on either line.

14. Two triangles ACB, ADB are constructed on the same side of the same base AB. Shew that if AC= BD and AD=BC, then CD is parallel to AB; but if_AC=BC_and AD=BD, then CD is perpendicular to AB.

15. AB is the hypotenuse of a right-angled triangle ABC: find a point D in AB such that DB may be equal to the perpendicular from D on AC.

16. Find the locus of the vertices of triangles of equal area on the same base.

17. Shew that the perimeter of an isosceles triangle is less than that of any triangle of equal area on the same base.

18. If each of the equal angles of an isosceles triangle be equal to one-fourth the vertical angle, and from one of them a perpendicular be drawn to the base, meeting the opposite side produced, then will the part produced, the perpendicular, and the remaining side, form an equilateral triangle.

19. If a straight line terminated by the sides of a triangle be bisected, shew that no other line terminated by the same two sides can be bisected in the same point.

20. From a given point draw to two parallel straight lines two equal straight lines at right angles to each other.

21. Given the lengths of the two diagonals of a rhombus, construct it.

22. ABCD is a quadrilateral figure: construct a triangle whose base shall be in the line AB, such that its altitude shall be equal to a given line, and its area equal to that of the quadrilateral.

23. If ABC be a triangle in which C is a right angle, shew how to draw a straight line parallel to a given straight line, so as to be terminated by CA and CB and bisected by AB.

24. If ABC be a triangle in which C is a right angle, and DE be drawn from a point D in AC at right angles to AB, prove that the rectangles AB, AE and AC, AD are equal.

25. A line is drawn bisecting a parallelogram ABCD and meeting AD, BC in E and F; shew that the triangles EBF, CED are equal.

26. Upon the hypotenuse BC and the sides CA, AB of a right-angled triangle ABC, squares BDEC, AF and AG are described: shew that the squares on DG and EF are together equal to five times the square on BC.

27. If from the vertical angle of a triangle three straight lines be drawn, one bisecting the angle, the second bisecting the base, and the third perpendicular to the base, shew that the first lies, both in position and magnitude, between the other two.

28. Shew that the area of a rhombus is equal to half the rectangle contained by the diagonals.

29. Let ACB, ADB be two right-angled triangles having a common hypotenuse AB. Join CD and on CD produced both ways draw perpendiculars AE, BF. Shew that the sum of the squares on CE and CF is equal to the sum of the squares on DE and DF.

30. In the base AC of a triangle take any point D: bisect AD, DC, AB, BC at the points E, F, G, H respectively. Shew that EG is equal and parallel to FH.

31. If AD be drawn from the vertex of an isosceles triangle ABC to a point D in the base, shew that the rectangle BD, DC is equal to the difference between the squares on AB and AD.

32. If in the sides of a square four points be taken at equal distances from the four angular points taken in order, the figure contained by the straight lines which join them shall also be a square.

33. If perpendiculars AP, BQ, CR be drawn from the angular points of a triangle ABC upon the sides, shew that they will bisect the angles of the triangle PQR.

34. If of the four triangles into which the diagonals divide a quadrilateral, any two opposite ones are equal, the quadrilateral is a trapezium.

35. ABCD, AECF are two parallelograms, EA, AD being in a straight line. Let FG drawn parallel to AC meet BA produced in G. Then the triangle ABE equals the triangle ADG.

36. From AC, the diagonal of a square ABCD, cut off AE equal to one-fourth of AC, and join BE, DE. Shew that the figure BADE is equal to twice the square on AE.

37. If ABC be a triangle with the angles at B and C each double of the angle at A, prove that the square on AB is equal to the square on BC together with the rectangle AB, BC.

38. If two sides of a quadrilateral be parallel, the triangle contained by either of the other sides and the two straight lines drawn from its extremities to the middle point of the opposite side is half the quadrilateral.

39. If two opposite angles of a quadrilateral be right angles, the angles subtended by either side at the two opposite angular points will be equal.

40. If the sides of a triangle taken in order be produced to twice their original lengths, and the outer extremities be joined, the triangle so formed will be seven times the original triangle.

41. If one of the acute angles of a right-angled isosceles triangle be bisected, the opposite side will be divided by the bisecting line into two parts, such that the square on one will be double of the square on the other.

42. ABC is a triangle, right-angled at B, and BD is drawn perpendicular to the base, and is produced to E until ECB is a right angle; prove that the square on BC is equal to the sum of the rectangles AD, DC and BD, DE.

43. Shew that the sum of the squares on two lines is greater than twice the rectangle contained by the lines.

44. In any triangle the sum of the squares on the straight lines, drawn from the angles to the middle points of the