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1867. III. 22. If the chords, which bisect two angles of a triangle inscribed in a circle, be equal, prove that either the angles are equal, or the third angle is equal to the angle of an equilateral triangle.

1868. 1. 41. OKBM and OLDN are parallelograms about the diameter of a parallelogram ABCD. In MN, which is parallel to BA, take any point P and prove that, if PC, produced if necessary, meet KL in Q, BP will be parallel to DQ.

II. 12. In a triangle ABC, D, E, F are the middle points of the sides BC, CA, AB respectively, and K, L, M are the feet of the perpendiculars on the same sides from the opposite angles. Prove that the greatest of the rectangles contained by BC and DK, CA and EL, AB and FM, is equal to the sum of the other two.

III. 35. Through a point within a circle, draw a chord, such that the rectangle contained by the whole

chord and one part may be equal to a given

square.

Determine the necessary limits to the magnitude of this square.

Iv. 4. If two triangles ABC, A'B'C' be inscribed in the same circle, so that AA' BB' CC' meet in one point 0, prove that, if O be the centre of the inscribed circle of one of the triangles, it will be the centre of the perpendiculars of the other.

1869. I. 40. ABC is a triangle, E and F are two points; if the sum of the triangles ABE and BCE be equal to the sum of the triangles ABF and BCF, then under certain conditions EF will be parallel to AC. Find these conditions, and determine when the difference instead of the sum of the triangles must be taken.

1869. II. 11. Shew that the point of section lies between the extremities of the line.

III. 33. An acute-angled triangle is inscribed in a circle, and the paper is folded along each of the sides of the triangle: Shew that the circumferences of the three segments will pass through the same point. State the equivalent proposition for an obtuse-angled triangle.

IV. 11. Shew that the circles, each of which touches two sides of a regular pentagon at the extremities of a third, meet in a point.

870. 1. 26. ABCD is a square and E a point in BC; a straight line EF is drawn at right angles to AE, and meets the straight line, which bisects the angle between CD and BC produced in a point F: prove that AE is equal to EF.

IL. 9. The diagonals of a quadrilateral meet in E, and F is the middle point of the straight line joining the middle points of the diagonals: prove that the sum of the squares on the straight lines joining E to the angular points of the quadrilateral is greater than the sum of the squares on the straight lines joining F to the same points by four times the square on EF.

я. 32. AB, CD are parallel diameters of two circles, and AC cuts the circles in P, Q: prove that the tangents to the circles at P, Q are parallel. rv. 10. Hence shew how to describe an equilateral and equiangular pentagon about a circle without first inscribing one.

1871. 1. 38. Through the angular points A, B, C, of a triangle are drawn three parallel straight lines meeting the opposite sides in A', B', C' respectively prove that the triangles AB'C', BCA', CA'B' are all equal.

II. 10. Produce a given straight line so that the square on the whole line thus produced may be double the square on the part produced.

1871. III. 32. The opposite sides of a quadrilateral inscribed in a circle are produced to meet in P, Q, and about the four triangles thus formed circles are described: prove that the tangents to these circles at P and Q form a quadrilateral equal in all respects to the original, and that the line joining the centres of the circles, about the two quadrilaterals, bisects PQ.

1872.

Iv. 5. A triangle is inscribed in a given circle so as to have its centre of perpendiculars at a given

point: prove that the middle points of its sides lie on a fixed circle.

1. 47. If CE, BD be the squares described upon the side AC, and the hypotenuse AB, and if

EB, CD intersect in F, prove that AF bisects the angle EFD.

II. 14. If the given rectilineal figure be that of Euclid 1. 47, shew how to determine the required

square graphically.

ПI. 22. Two circles intersect in A, B: PAP', QAQ' are drawn equally inclined to AB to meet the

circles in P, P', Q, Q: prove that PP' is equal to QQ'.

17. 4. Having given an angular point of a triangle, the circumscribed circle, and the centre of the inscribed circle, construct the triangle.

BOOK V.

SECTION I.

On Multiples and Equimultiples.

DEF. I. A GREATER magnitude is a Multiple of a less magnitude, when the greater contains the less an exact number of times.

DEF. II. A LESS magnitude is a Sub-multiple of a greater magnitude, when the less is contained an exact number of times in the greater.

These definitions are applicable not merely to Geometrical magnitudes, such as Lines, Angles, and Triangles; but also to such as are included in the ordinary sense of the word Magnitude, that is, anything which is made up of parts like itself, such as a Distance, a Weight, or a Sum of Money.

POSTULATE.

Any one magnitude being given, let it be granted that any number of other magnitudes may be found, each of which is equal to the first.

METHOD OF NOTATION.

Let A represent a magnitude, not as one of the letters used in Algebra to represent the measure of a magnitude, but let A stand for the magnitude itself. Thus, if we regard A as representing a weight, we mean, not the number of pounds contained in the weight, but the weight itself.

Let the words A, B together represent the magnitude obtained by putting the magnitude B to the magnitude A.

Let A, A together be abbreviated into 24,

A, A, A together

and so on.

Let A, A.

..repeated m times be denoted by mA,

m standing for a whole number.

Let mA, mA......repeated ʼn times be denoted by nmA, where nm stands for the arithmetical product of the whole numbers n and m.

Let (m+n) A stand for the magnitude obtained by putting nA to mA, m and n standing for whole numbers.

These, and these only, are the symbols by which we propose to shorten and simplify the proofs of this Book: capital letters standing, in all cases, for magnitudes; and small letters standing for whole numbers.

SCALES OF MULTIPLES.

By taking a number of magnitudes each equal to A, and putting two, three, four......of them together, we obtain a set of magnitudes, depending upon A, and all known when A is known; namely,

A, 2A, 3A, 4A, 54............and so on;

each being obtained by putting A to the preceding one.

This we call the SCALE OF MULTIPLES of A.

If m be a whole number, mA and mB are called Equimultiples of A and B, or, the same multiples of A and B respectively.

AXIOMS.

1. Equimultiples of the same, or of equal magnitudes, are equal to one another.

2. Those magnitudes, of which the same, or equal, magnitudes are equimultiples, are equal to one another.

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