62. If a parallel to the side AB, of a triangle ABC, meets CA in X, and CB in Y; then the Locus of the intersection of the circles round the triangles CAY, CBX, is a line through C.

63. If a corner of a triangle is joined to the point of contact of the in-circle (or an ex-circle) with the side opposite; then the mid point of this join, and the mid point of the side, are collinear with the centre of the circle.

64. In the figure of vi. Addenda (7) if the triangle varies subject to the conditions that BC is fixed, and BA + AC is constant; then AX: XD is constant. 65. In Exercise 100, page 188, show that OA: OX OY: OC.

=

66. In a triangle ABC, M is the mid point of BC, D the point where the bisector of angle BAC (internal or external) meets BC; then, if MX, BY, CZ, are perpendiculars on the other bisector (external or internal) of angle BAC-BY.CZ MX. AD.

67. Four rods PA, PB, QAC, QBD are pivoted at P, Q, A, B, so as to be capable of angular motion in one plane; and so that PAQB is a parallelogram: if any pair of fixed points in QC, QD, respectively, are once collinear with P, they will always be so, however the rods are moved about.

NOTE-This is virtually the same as the omitted vi. 32. Ex. 62, p. 82, is a particular case of it.

68. Two triangles are similar and similarly situated; if a third triangle can be drawn to circumscribe the inner and be inscribed in the outer, then its area is a mean proportional between the areas of the original triangles.

NOTE-Take the centre of similarity of the ▲s: see vi. Addenda (3).

69. ABCD is a quadrilateral circumscribing a circle (centre O) and XOY is perpendicular to the bisector of the angle between BA, CD-X being in AB, and Y in CD-show that AX: BX CY: DY.

NOTE-Use Exercise 41.

=

70. Prove Exercise 113, p. 190, by vi. Addenda (24) and the last Exercise. NOTE-This is Newton's original mode of proof.

71. ABCD is any quadrilateral; and M, N are the respective mid points of AC, BD: if MN produced meets AB, BC, CD, DA in P, Q, R, S respectively, then

And if the quadrilateral can have a circle (centre O) inscribed in it, then

*

72. The following group are developments of Note (4) p. 253(1) If we take two maps of the same country, on different scales, and throw one on the other, there will be one spot (and only one) whose position on the one map will be exactly over its position on the other; provided that the contour of the lesser map is wholly within that of the larger.

Also, if the maps are not superposed, but simply laid at random on a table, there will be one point on the table which will represent the same place to whichever map it may be considered to belong.

(2) If corresponding points are taken, one on each map; then the Locus of the point of intersection of corresponding lines through them is a circle.

(3) If a series of parallels is taken on one map; then the Locus of the intersection of each with the corresponding line in the other map, is a straight line.

(4) If a series of concentric circles is taken in one map; then the Locus of the intersection of each with the corresponding circle in the other map is a circle.

NOTE-Use vi. Addenda (15).

(5) Two corresponding points, one on each map, are held fixed, while the maps are moved about; find the Locus of the centre of similarity.

(6) A pin is put through both maps at a given point; find the Locus of the centre of similarity, as one or other map is turned round.

(7) Find the Loci of corresponding points, one on each map, whose distance apart is constant.

(8) Find the Envelope (cf. Note on Exercise 59) of the joins of the points in (7).

(9) If in (5) any two corresponding points, which are at a given distance apart, are fixed, we get a series of Loci; find their Envelope.

(10) Find the Envelope of the Loci in (2), when the points are at a given distance apart.

(11) If a circular disc is placed anywhere on the maps, it must cover at least some corresponding points, provided its centre lies within a certain circle; otherwise it will not cover any corresponding points.

* Of this group (1) and (7) are due to Professor Purser, of Queen's College, Belfast; and the rest to Mr. Alexander Larmor, of Clare College, Cambridge.

GENERAL ADDENDA.

SECTION i—MAXIMA AND MINIMA.

Def. If a straight line, or angle, or area, can vary, subject to given limitations; it is said to be maximum, when it has its greatest possible value; and minimum, when it has its least possible value.

Some cases of maxima and minima have already occurred: see iii. 7, 8, 15; i. Addenda (2); iii. Addenda (2); vi. Addenda (9). Here follow some Theorems which may be regarded as fundamental.

THEOREM (1)—The sum of the squares on the two segments into which a given line can be divided, is minimum, when the line is bisected.

THEOREM (2)-The rectangle under the two segments into which a given line can be divided, is maximum when the line is bisected.

Cor. Of all rectangles, of given perimeter, the square has the maximum area.

THEOREM (3)-If the rectangle under two lines is given, the sum of the lines is minimum when they are equal.

Follows at once from ii. Addenda (3).

Cor. Of all rectangles, of given area, the square has the minimum perimeter.

THEOREM (4)—If the sum of the squares on two lines is given, the sum of the lines is maximum when they are equal.

Follows at once from ii. Addenda (6).

Note-When any two magnitudes whatever are commensurable (so that they can be expressed by x and y units of measurement respectively), Theorems, analogous to the foregoing, are seen to be true, from the two algebraic identities

And similar theorems will follow, for the reciprocals of the magnitudes, from the identities

Def. When two magnitudes are so related that they vanish together, and that equal increments of the one involve equal increments of the other, the magnitudes are said to vary one as the other: such magnitudes will be maxima together, and minima together.

THEOREM (5)—The maximum parallelogram which can be inscribed in a triangle, by drawing parallels to two of its sides, is that formed by drawing the parallels from the mid point of the third side; and its area is half that of the triangle.

B

Let PX, PY be Is to sides AB, AC, of ▲ ABC, drawn from pt. P in BC.

and this, by Theor. (2) is max. when P is mid pt. BC.
In that case X, Y are also mid pts. of AC, AB;
and.. AP = { ▲ ABC.

Note-The preceding Theorem is practically equivalent to the omitted vi. 27.

THEOREM (6)—-The maximum triangle which can be inscribed in a given segment of a circle, is that formed by joining the mid point of its arc to the extremities of its chord.

Cor. If the base and vertical angle of a triangle are given, the triangle is maximum when it is isosceles.

THEOREM (7) When two sides of a triangle are given in length, the area of the triangle is maximum when they are placed at right angles.