Ex. 22. ABCD, ABXY are two parallelograms on the same base and on the same side of it. Prove that CDYX is a parallelogram.

Ex. 23. ABCD is a parallelogram; DA and DC are produced to X and Y respectively so that AXDA and CY= DC; XB and BY are drawn. Prove that XBY is a straight line.

Ex. 24. In a triangle ABC, BE and CF are drawn to cut the opposite sides in E and F; prove that BE and CF cannot bisect one another.

Ex. 25. Show how to place a straight line of given length with one end on each of two intersecting lines and parallel to another straight line.

Ex. 26. Show how to construct a line such that it is divided into two equal parts by three intersecting straight lines.

Isosceles Trapezium*.

Ex. 27. Prove that a line drawn parallel to the base of an isosceles triangle to cut the sides divides the triangle into an isosceles triangle and an isosceles trapezium.

Ex. 28. ABCD is an isosceles trapezium (AD=BC); prove that C=LD. [Through B draw a parallel to AD.]

Ex. 29. If in Ex. 28 E, F are the mid-points of AB, CD, then EF is perpendicular to AB. [Join AF, BF.]

Ex. 30. In a quadrilateral ABCD, AB=CD and ▲ B= ≤C; prove that AD is parallel to BC.

Ex. 31. Prove that the diagonals of an isosceles trapezium are equal. Ex. 32. ABCD is a quadrilateral, such that ▲ A B and ▲ C=D; prove that AD BC.

=

MID-POINT AND INTERCEPT THEOREMS.

Ex. 33. Show how to cut off from a given straight line any fraction (say) of its length.

Ex. 34. X, Y, Z are the mid-points of the sides BC, CA, AB of a ▲ ABC. Prove that AZXY is a parallelogram.

Ex. 35. The straight lines joining the mid-points of the sides of a triangle divide it into four congruent triangles.

Ex. 36. Given the three mid-points of the sides of a triangle, construct the triangle. Give a proof.

Ex. 37. If D, E are points in AB, AC such that AD=AB and AE=1AC, prove that DE is parallel to BC and equal to a quarter of BC.

Ex. 38. T, V are the mid-points of the opposite sides PQ, RS of a parallelogram PQRS. Prove that ST, QV trisect PR.

Ex. 39. Prove that the median of a triangle bisects the line joining the mid-points of the other two sides of the triangle.

*For definition see Index.

Ex. 40. Any straight line drawn from the vertex to the base of a triangle is bisected by the line joining the mid-points of the sides.

Ex. 41. Through D, the mid-point of the base BC of a triangle ABC, lines are drawn parallel to the sides of the triangle to cut the sides at E and F. Prove that EF=BC.

Ex. 42. The medians BE, CF of a triangle ABC intersect at G; GB, GC are bisected at H, K respectively. Prove that HKEF is a parallelogram. Hence prove that G is a point of trisection of BE and CF.

Ex. 43. C is the mid-point of AB; from A, B, C perpendiculars AX, BY, CZ are drawn to a given straight line. Prove that, if A and B are both on the same side of the line, AX + BY=2CZ. [Join AY.]

What relation is there between AX, BY, CZ when A and B are on opposite sides of the line?

If a parallelogram is held under water, show that in every position the sum of the depths of the four corners is 4 times the depth of the point of intersection of the diagonals.

Ex. 44. If the mid-points of the adjacent sides of a quadrilateral are joined, the figure thus formed is a parallelogram.

[Draw a diagonal of the quadrilateral.]

Ex. 45. The straight lines joining the mid-points of opposite sides of a quadrilateral bisect one another.

Ex. 46. Are Exx. 44, 45 true for a skew quadrilateral (see p. 42) ?

Ex. 47. In a right-angled triangle, the distance of the vertex from the mid-point of the hypotenuse is equal to half the hypotenuse.

[Join the mid-point of the hypotenuse to the mid-point of one of the sides.] Ex. 48. Given in position the right angle of a right-angled triangle and the length of the hypotenuse, find the locus of the mid-point of the hypotenuse, (See Ex. 47.)

CHAPTER VII

AREA

The number of square units in the area of a rectangle is found by multiplying together the numbers of units in the length and breadth of the rectangle.

This has been proved in Practical Geometry, Chap. 7, for the case in which the sides of the rectangle contain exact numbers of units of length. The method there used can be applied to any required degree of accuracy by taking a sufficiently small unit of length, but the general proof involves very considerable difficulties concerned with incommensurables*.

Euclid's attitude towards area is discussed in Chap. VIII, p. 61. Figures which are equal in area are said to be equivalent. THEOREM 25. Parallelograms on the same base and between the same parallels (or, of the same altitude) are equivalent.

PA Q D P

B

Fig. 46.

DP

QA

Data ABCD, PBCQ are ||ograms on the same base BC, and between the same parallels BC, PD.

To prove that ABCD and PBCQ are equivalent.

[These figures are left if As PBA, QCD are taken from PBCD.]

Proof In the As PBA, QCD,

Now if PBA is subtracted from figure PBCD, ||ogram ABCD is left; and if ▲ QCD is subtracted from figure PBCD, ||ogram PBCQ is left. Hence the lograms ABCD, PBCQ are equivalent.

Q. E. D.

*Further information may be found in Dedekind's Essays on the Theory of Number (Open Court Publishing Co.).

COR. 1. A parallelogram is equivalent to the rectangle on the same base and between the same parallels (or, of the same altitude); and the area of a parallelogram is measured by the product of the base and the altitude.

This is a special case of the theorem.

The same proof as for the theorem applies to the special case. COR. 2. Parallelograms on equal bases and of the same altitude are equivalent.

(For they can be so placed as to be on the same base and between the same parallels.)

Ex. 1. Write out the proof of Cor. 1.

THEOREM 26. Triangles on the same base and between the same parallels (or, of the same altitude) are equivalent.

B

A

Fig. 47.

Data ABC, PBC are As on the same base BC, and between the same parallels BC, PA.

To

prove that

ABC, PBC are equivalent.

Construction Complete the [ograms ABCD, PBCQ by drawing CD, CQ || to BA, BP respectively, to meet PA (produced if necessary) in

But ||ograms ABCD, PBCQ are equivalent, being on the same base and between the same parallels.

.. AABC = A PBC.

Th. 25.

Q. E. D.

COR. 1. The area of a triangle is half the area of the rectangle on the same base and between the same parallels (or, of the same altitude), and is measured by half the product of the base and the altitude.

COR. 2. Triangles on equal bases and between the same parallels

are equivalent.

Ex. 2. Prove Cor. 1 for the triangle ABC when LABC is a right angle.

Ex. 3. Prove Cor. 1 without using a parallelogram by dividing the triangle into two right-angled triangles.

THEOREM 27. Equivalent triangles which have equal bases in the same straight line, and are on the same side of it, are between the same parallels.

C

B

Fig. 48.

Data ABC, DEF are equivalent triangles on equal bases AB, DE, these being in a straight line, and C and F being on the same side of AE.

If possible, draw a line CG || to AE, distinct from CF, meeting

FD (produced if necessary) in G. Join EG.

Proof [We shall prove that CG coincides with CF.]

Since AB = DE, and CG is || to AE,

.. AABC=▲ DEG.

But AABC A DEF,

=

.. A DEF = A DEG,

.. F coincides with G, and CF with CG,

.. CF is || to AE.

Th. 26, Cor. 2.

Data.

Q. E. D.

COR. 1. Equivalent triangles on the same or equal bases are of the same altitude.

COR. 2. Equivalent triangles on the same base and on the same side of it are between the same parallels.

COR. 3. The locus of the vertices of equivalent triangles on the same base and on the same side of it is an unlimited straight line parallel to the base.

Ex. 4. Give another type of proof of Cor. 1.