.. DK=BM+BL and AL= CM.

.. MOOL and A0=00. [I. 26.]

.. DK=BM+BL=2BO and AC is bisected.

.. DK-2 the bisector BO. Similarly, if AR is the bisector of BC; EG=2AR and FH=2CS. And since DK, EG and FH are known the three bisectors of the triangle ABC are known, and the triangle may now be constructed by Ex. 183.

(198.) Analysis. If AB is the given base and O its middle point, join OC. Then Clies in the circumference of a circle drawn from O with radius = OC.

Again, let BH be the perpendicular on AC. Join OH. Then OH=0A=OB. Ex. 4, prop. 32, School Euclid. H lies in a semicircle described from O with radius = OA.

But I also lies in a circle described from B with the given length BH as radius.

The intersection of these two latter loci determine the point H. If then AH be joined and produced to cut the circle described from O with radius = OC the point will be determined.

(199.) Analysis. Let ABC be the triangle, and FG the line drawn parallel to AB, so that FG = AF+BG.

Make FO=FA; then GO=GB. Join OA, OB.

Then the FAO, FOA are equal. [I. 5.]

But the 4o FOA = 2o OAB. [I. 29.]

.. The CAB is bisected by AO.

is bisected by BO.

Similarly, OB,

The intersection of these lines gives the point 0, through which the parallel FG is drawn.

A similar analysis applics to the remaining case.

to AE meeting these perpendiculars in B and C. Then AC is a rectangle. Draw GD and CL parallel to RE and PQ, and prove the triangles BLC, GCD equal by I. 34 and I. 26.

.. BC=CD. Hence the fig. is equilateral.

.. It is a square.

103

APPENDIX TO BOOK ii.

The following arrangement of the first ten propositions in the second book of Euclid presents the results in their more essential connection.

Props. 2 and 3. These propositions may be deduced as corolloraries to proposition 1, of which they are only particular instances.

Prop. 4.-Next, proposition four may be deduced from the results already obtained, as shewn on page 87 of the School Euclid.

The result of prop. 4 may be thus stated,

The square on the sum of two lines is greater than the squares on the two lines by twice the rectangle contained by them.

Prop. 7 should follow proposition 4, and it may be stated thus:

The square on the difference of two lines is less than the squares on the lines by twice the rectangle contained by them.

That this simply presents the same proposition in another form may be seen by referring to the demonstration on page 90 of the Euclid, where the sq. on AC (which is the difference of AB and BC) is proved to be less than the sqs. on AB, and BC by twice the rectangle AB, BC.

Prop. 8 follows from props. 4 and 7 by subtraction. For referring to the fig. on page 92 of the Euclid, AD is evidently the sum of AB, and BC and AC their difference. Hence, since BC=BD.

The sq. on AD

=

sq. on AB + sq. on BC + 2 rect.

sq. on AC sq. on AB + sq. on BC

AB.BC. [ii. 4.]

AB.BC. [ii. 7.]

2 rect.

the sq. on AC = 4 rect. AB.BC.

[ii. 8.]

The result agrees with prop. 8, and may be thus stated:

The square on the sum of two unequal lines exceeds the square on their difference by four times the rectangle contained by them.

Props. 9 and 10 may in like manner be derived from props. 4 and 7 by addition.

For, looking to the diagrams on pp. 94 and 96, it is plain that AD is the sum of half the line (viz. AC), and the line between the points of section (viz. CD), and DB is the difference of the same two lines.

. The sq. on AD = sq. on AC + sq. on CD + 2 rect. AC.CD. [ii. 4.]

the sq. on DB = sq. on AC + sq. on CD 2 rect. AC.CD. [ii. 7.]

. The sq. on AD + sq. on DB 2 sq. on CD. [ii. 9 and 10.]

The result shews that

= 2 sq. on AC +

The square on the sum of two unequal lines increased by the square on their difference is equal to twice the squares on the lines.

Props. 5 and 6 may be demonstrated as in the text. But since the greater of the unequal segments in each case is the sum of half the line, and the line between the points of section C and D, and the less unequal segment is the difference of the same two lines, these propositions may be included in one statement, viz.:

The rectangle contained by the sum and difference of two unequal lines is equal to the difference of the squares described on the lines.

The principal results therefore, five in number, are contained in props. 4, 7, 8, (9 and 10), (5 and 6), and they may be presented symbolically thus:

1. (a+b)2=a2+b2+2ab.

2. (a-b)2=a2+b2-2ab.

3. (a+b)-(a—b)2=4ab.

4. (a+b)2+(a−b)2=2a2+2b2.

5. (a+b)(a−b) = a2 — b2.

[ii. 4.]

Tii. 7.1

ii. 8.]

Tii. 9 and 10.]

[ii. 5 and 6.]