EXERCISES ON BOOK II.

1. In a right-angled triangle, the square upon the greater of the sides containing the right angle, is equal to the rectangle contained by the sum and difference of the hypotenuse and the other side..

2. The square described upon the difference of two straight lines is equal to the sum of the squares described on the two lines diminished by twice their rectangle.

3. The sum of the squares on two lines is equal to twice the square on half their sum, together with twice the square on half their difference.

4. The square on the perpendicular from the right angle to the hypotenuse, is equal to the rectangle contained by the segments of the hypotenuse; and the square on each side is equal to the rectangle of the hypotenuse, and the segment of it coterminous with the side.

5. ABC is an equilateral triangle, AC is bisected in D, DE drawn perpendicular to BC and BD joined; shew that the square on BD is three-fourths of the square on BC, and the part BE three-fourths of BC.

6. If from the angles of a triangle lines be drawn to the middle of the opposite sides, four times the sum of the squares on these lines is equal to three times the sum of the squares on the sides of the triangle.

7. In any quadrilateral the squares on the diagonals are together equal to twice the sum of the squares on the straight lines joining the middle points of the opposite sides.

8. If two sides of a trapezium be parallel, the sum of the squares on the diagonals will be together equal to the sum of the squares on the parallel sides, together with twice the rectangle contained by these sides.

9. In a right-angled triangle ABC, of which C is the right angle, if DE is drawn from a point D in AC perpedicular to AB, the rectangle contained by AB and AE is equal to the rectangle contained by AC and AD.

10. If from the middle point of one of the sides of a right-angled triangle a perpendicular be drawn to the hypotenuse, the square on the third side is equal to the difference of the squares on the segments of the hypotenuse.

11. If perpendiculars be drawn from the three angles of a triangle to the opposite sides, the sums of the squares on the alternate segments of the sides are equal to one another.

12. Divide a straight line into two parts, so that the rectangle contained by the two parts shall be equal to the square on a given line. How many solutions are there? When is the problem impossible?

13. Produce a given line, so that the rectangle contained by the whole line thus produced, and the part produced, shall be equal to a given square.

14. Divide a line into two parts, so that the sum of the squares on the parts shall be equal to a given square. How many solutions are there? When is the problem impossible?

15. Divide a line into two parts, so that the difference of the squares on the parts shall be equal to a given square. When is the problem impossible?

16. Produce a given line, so that the rectangle contained by the whole line thus produced, and the given line, may be equal to a given square.

17. The square on the base of an isosceles triangle is equal to twice the rectangle contained by either of the equal sides, and the projection of the base upon that side.

18. The square on the hypotenuse of a right-angled triangle is equal to four times the area of the triangle, together with the square on the difference of the two sides.

19. Produce a given straight line, so that the square on the whole line thus produced shall be equal to twice the square on the given line. 20. ABC is an obtuse angled-triangle, having C obtuse; if AQ and BO be let fall from the angles A and B on BC and AC produced, then shall the rectangle contained by AC and CO be equal to that contained by BC and CQ.

21. If an angle of a triangle be double the angle of an equilateral triangle, the square on the side subtending that angle is equal to the sum of the squares on the sides which contain it, together with the rectangle contained by these sides.

22. Any rectangle is half the rectangle contained by the diagonals of the squares described upon its two sides.

23. If a straight line be drawn from one of the acute angles of a right-angled triangle to the point of bisection of the opposite side, the square on the hypotenuse is greater than the square on the line so drawn by three times the square on half the side bisected.

24. If the straight line AB be divided unequally in C, and equally in D, the squares on AC and CB are greater than twice the rectangle contained by AC and CB, by four times the square on the intercept CD.

25. Let CAB be a right-angled triangle, right-angled at C, and let the square ABKO be described on the hypotenuse AB; then if CO and CK be drawn to the remote angles O and K of the square, the difference of the squares on these two lines will be equal to the difference of the squares on the sides AC and CB of the triangle.

26. If the hypotenuse AB of a right-angled triangle be trisected in the points O and Q, and these points joined to the right angle C, then shall the sum of the squares on the sides of the triangle COQ be equal to two-thirds of the square on the hypotenuse AB.

27. In a straight line divided as in II. 11, the rectangle contained by the sum and difference of the parts is equal to the rectangle contained by the parts.

28. If a straight line DE be drawn parallel to the base BC of an isosceles triangle ABC, and the points B, E be joined, the square on BE shall be equal to the rectangle contained by BC and DE, together with the square on CE,

BOOK III.

DEFINITIONS.

1. A secant is a straight line of unlimited length, which meets the circumference of a circle in two points.

line PQ, which meets the circum

ference in the points A and B, and is said to cut the circle, is a secant.

2. A tangent is a straight line of unlimited length, which meets the circumference of a circle, and being produced does not cut the circle. Thus if we suppose the point B to approach nearer and nearer to A, and ultimately_to coincide with it, the secant PQ

P

P

Thus the

B

B

B

will successively take the positions P'Q', P"Q", and ultimately the definite position CAD, which is the tangent at the point A. This point, which is common to the tangent and the circumference, is the point of contact, or point in which the straight line touches the circle.

B

3. Two circumferences are said to cut one another when

they have two points in common.

When the two points are

coincident, the circumferences are said to touch one another; and the common point is the point of contact or of tangency. Two kinds of tangency are distinguished: external, when each circle is without the other; and internal, when one circle is within the other.

S

A

B

4. A locus is an assemblage of points which have a common property. Thus the straight line AB is the locus of all the points O, P, Q, which are equally distant from the extremities of the line CD; also the circumference of a circle is the locus of all the points which are equally distant from a given point.

PROPOSITIONS.

PROPOSITION I.-THEOREM. (Euc. III. 1, cor.)

The centre of a circle lies on the straight line which is drawn at right angles to any chord through its middle point.

Let AB be any chord of the circle ABE, and let CE be a line drawn through its middle point C, at right angles to AB; it is required to prove that the centre of the circle is in the perpendicular CE. If the centre be not in CE, let it be at any other point D.

B

Join DA, DC, and DB.

Then in the two triangles DCA and DCB, we have

AC equal to BC, and CD common to both triangles, and the third side DA in the one

equal to the third side DB in the other,

each being supposed to be a radius;

therefore the angle ACD is equal to the angle BCD, and therefore each of them is a right angle.

(I. Def. 30.)

(I. 6.) (I. Def. 8.)

But ACE is a right angle;

therefore the angle ACE is equal to the angle ACD,

the part to the whole, which is absurd.

(Hyp.)

(Ax. 9.)

Therefore the centre of the circle cannot lie out of the line drawn through the middle point of the chord at right angles to it. Q. E. D.

E

E

B

Cor. 1. (Euc. III. 1).—Hence the centre of a circle will be found by drawing two chords AB and A'B', which are not parallel, and raising perpendiculars CE and C'E' from their middle points C and C'; the intersection O of the perpendiculars determines the centre. the centre must lie on both lines, and therefore be their point of intersection.

For A

C

B

Cor. 2.-It follows that a circle may be described passing through three points, when these are not in the same straight line.

Conversely (Euc. III. 3).-1. A line which passes through the centre of a circle, and is at right angles to a chord, bisects it. Let the line EC pass through the

centre O of the circle ABF, and be at right angles to the chord AB, it shall pass through the middle point C of the chord.

Join O, A and O, B.

Then in the two triangles OCA, A

OCB, we have

The two angles OAC, OCA of the one triangle,

C

equal to the two angles OBC, OCB respectively, of the other triangle, and the side OC common to both triangles; therefore the side AC is equal to the side BC, and AB is bisected in C. Q. E. D.

(I. 19.)

2. A line which passes through the centre of a circle, and bisects a chord which does not pass through the centre, is at right angles to the chord.

Let the line EC pass through the centre O of the circle ABF, and through C the middle point of the chord AB,