draw GM parallel to AB, and KN to CD,

meeting BH and DL produced if necessary, at M and N respectively.

CD,

Then because GM is parallel to AB, KN to CD, and AB to

therefore GM is parallel to KN,

I. 24.

I. 23, Cor. 2.

therefore the angle MGH is equal to the angle NKL.

Hence in the triangles MGH, NKL,

the angle MGH is equal to the angle NKL,

and the angle GHM is equal to the angle KLN, each being a

THEOR. 33. Equal straight lines that have equal projections on another straight line are parallel to that line, or make equal angles with it.

Let AB, CD be two equal straight lines, GH, KL their

projections on another straight line EF, and let GH be equal to KL:

then shall AB and CD be parallel to EF, or make equal angles with EF.

EF,

If either AB or CD is parallel to EF, let AB be parallel to

H

then AGHB is a parallelogram,

and therefore AB is equal to its projection GH, therefore CD is also equal to its projection KL;

but the perpendicular from C upon DL is equal to KL,

therefore CD is this perpendicular,

and therefore CD is parallel to EF.

I. 29.

Hyp.

I. 29.

I. 15.

Therefore AB and CD are both parallel to EF if either of them is.

But if neither AB nor CD is parallel to EF,

draw GM parallel to AB, and KN to CD, meeting BH and DL produced if necessary at M and N respectively,

let AB, CD produced if necessary meet EF at O and P respectively.

Then since AB is equal to CD

therefore GM is equal to KN.

Hence in the right-angled triangles MGH, NKL,

the side GM is equal to the side KN,

and the side GH is equal to the side KL,

Hyp. I. 29.

Hyp.

therefore the angle MGH is equal to the angle NKL, I. 20, Cor. 1. therefore the angle AOG is equal to the angle CPK, I. 23, Cor. 2. that is, AB and CD make equal angles with EF.

Q.E.D.

THEOR. 34. If there are two pairs of straight lines all of which are parallel, and the intercepts made by each pair on a straight line that cuts them are equal, then the intercepts on any other straight line that cuts them are also equal.

Let AB, CD and EF, GH be two pairs of straight lines all of which are parallel, and let the intercepts AC, EG on the straight line AG be equal:

K

E

H

then shall the intercepts BD, FH on any other straight line BH

which cuts them be equal.

If BH is parallel to AG,

then BD is equal to AC,

1. 29.

and FH is equal to EG,

I. 29.

therefore BD is equal to FH.

But if BH is not parallel to AG,

draw BK and FL parallel to AG, meeting CD and FH in K and

L respectively,

then BK is equal to AC,

and FL is equal to EG,

1. 29.

1. 29.

therefore BK is equal to FL.

Hence in the triangles BKD, FLH

the angle KBD is equal to the angle LFH,

the angle BDK is equal to the angle FHL,

and the side BK is equal to the side FL, therefore the side BD is equal to the side FH.

COR. 1. If there are three parallel straight lines, and the intercepts made by them on any straight line that cuts them

are equal, then the intercepts on any other straight line that cuts them are also equal.

COR. 2. The straight line drawn through the middle point of one of the sides of a triangle parallel to the base passes through the middle point of the other side.

COR. 3. The straight line joining the middle points of two sides of a triangle is parallel to the base.

Ex. 58. The straight lines joining the middle points of the sides of a triangle divide it into four identically equal triangles.

*Ex. 59. The straight line joining the middle points of two sides of a triangle is equal to half the base.

Ex. 6o. If a quadrilateral be formed by joining the middle points of the sides of a given quadrilateral, it is a parallelogram.

EXERCISES

61. Prove that the angle between the bisectors of two adjoining angles of a quadrilateral is half the sum of the two remaining angles.

62. If the sides of a regular pentagon be produced to meet, the angles formed by these lines are together equal to two right angles.

63. If a straight line parallel to BC, the base of an isosceles triangle ABC, meet the sides AB, AC at D and E: shew that the triangles CDE, DCB have two sides and one angle of the one equal to two sides and one angle of the other. Are they equal in all respects?

64. Straight lines AD, BE, CF are drawn within the triangle ABC, making the angles DAB, EBC, FCA all equal to one another. If the lines AD, BE, CF do not meet at a point: prove that the angles of the triangle formed by them are equal to those of the triangle ABC, each to each.

65. The exterior angles at B and C of the triangle ABC are bisected by lines meeting at D. Shew that the angle BDC is equal to half the exterior angle at A.

66. If a quadrilateral has two sides parallel, and the other two sides equal but not parallel, shew that the diagonals of the quadrilateral are equal.

67. Two triangles ABC, DBC are upon the same base BC, and AD is parallel to BC. If ABC is isosceles, shew that its perimeter is less than that of DBC.

68. The sum of the distances of any point in the base of an isosceles triangle from the two sides is constant.

E