Proposition 10.

Theorem.-The sum of the squares of the diagonals of any parallelogram, is equal to the sum of the squares of the sides of the parallelogram.

Let ABCD be a parallelogram, and AC, BD its diagonals;

then

AC2 + BD2 = AB2 + BC2 + CD2 + DA2.

Let AC and BD intersect each other in E; and because the vertical angles AED, CEB are equal (I. 17), and also the alternate angles EAD, ECB (I. 27), the triangles AED, CEB have two

A

B

angles of each equal, and the sides AD, BC are equal (I. 32); therefore the other sides, which are opposite the equal angles are also equal—namely, AE to EC, and DE to EB. Since, therefore, DB is bisected in E from (II. 9),

AD2+AB2 = 2DE2+2AE'.

For the same reason,

DC2+ CB22DE2+2EC2 = 2DE2+2AE2;

AD2 + AB2 + DC2 + CB2 = 4DE2+4AE2.

But 4DE2 DB2 (II. 4, Cor. 2), and 4AE2 = AC2;

=

AD2+AB2 + DC2 + CB2 = DB2 + AC2.

Corollary.-From this demonstration it is manifest that the diagonals of a parallelogram bisect each other.

Proposition 11.

Problem.—To divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts, may be equal to the square of the other part.

Let AB be a given straight line. It is required to find the point H in it, so that

Now (I.42), EB2 = AE2+ AB2; .. CF. FA+AE2 = AE2 + EB2. Taking away the common AE, and

Now,

and

CF. FA-AB2.

CF. FA is the rectangle FK,

AB is the square AD.

Therefore the rectangle FK is equal to the square AD. Taking away the common rectangle AK, the remainder FH is equal to the remainder HD.

But

HD=DB.BH = AB.BH,

and FH-AH';

... AB.BH = AH', and AB is divided in H, so that the rectangle contained by the whole and one part is equal to the square of the other.

Proposition 12.

Problem.-To describe a square that shall be equal to a given polygon.

Let A be a given polygon; it is required to describe a square that shall be equal to it.

Describe (I. 39, Sch.), the rectangle BCDE equal to A. If the sides BE, ED be equal to each other, the thing required is done; but if not, produce one of them, BE to F, and make EF equal

to ED, and bisect (I. 10) BF in G; on BF describe the semicircle BHF; produce DE to meet it in H, and join GH.

B

A

E

F

Because BE is the sum and EF the difference of FG and

But BD is the rectangle BE. ED or BE. EF, and BD is equal by construction to A. Therefore the square on EH is equal to the rectilineal figure A.

EXERCISES.

1. The diagonals of a rhombus bisect each other at right angles.

2. Two parallelograms are equal when they have two sides and the included angle, each to each.

3. The sum of the diagonals of a trapezium is less than the sum of the four lines drawn from any point within the figure to the four angles.

4. If AD be a square, and CE be cut off its diagonal equal to AB, and EF be drawn at right angles to CB, then BE- EF = FD.

5. The squares of the diagonals of a tra

A

B

C

D

pezium are together less than the squares of the four sides, by four times the square of the line joining the middle points the diagonals.

B

6. If a straight line be divided into two equal, and also into two unequal, parts, the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line.

If AB be divided equally at C and unequally at D, then

AD.DB+CD2 = CB2.

C

Describe a square on CB, and proceed as in (II. 6).

D B

7. Divide a given straight line into two parts so that the rectangle contained by them shall be the greatest possible. Use the property of last exercise, and show that the middle point is the point of division.

8. Construct a rectangle equal to the difference between two given squares.

From Ex. 6 we have AD. DB = CB2 - CD. From this the construction may be deduced.

9. The centre of a circle is on the middle point of the base of a triangle; if the vertex be any point of the circumference, the sum of the squares of the two sides is equal to the square of the diameter (II. 9).

10. The square on the perpendicular, drawn from the right angle of a right-angled triangle to the base, is equal to the rectangle contained by the segments of the base.

Bisect the base, and use Exs. 6 and 9.

BOOK III.

CIRCLES.

DEFINITIONS.

1. A STRAIGHT line is said to touch a circle when it meets the circle, and being produced does not cut it.

And that line which has but one point in common with the circle is called the tangent, and the point in common, the point of contact.

2. Circles are said to touch each other when they meet, but do not cut.

3. Straight lines are said to be equally distant from the centre of a circle, when the perpendiculars drawn to them from the centre are equal;

TANGENT

4. And the straight line on which the greater perpendicular falls, is said to be farther from the centre.

5. An arc is any portion of the circumference.

6. The chord of an arc is the straight line which joins its extremities.

7. The segment of a circle is the portion cut off by a chord.