of one of these is equal to the sum of the latera-recta of the other three.

3. On a fixed tangent to a conic are taken a fixed point A, and two moveable points P, Q, such that AP, AQ, subtend equal angles at a fixed point 0. From P, Q are drawn two other tangents to the conic, prove that the locus of their point of intersection is a straight line.

4. Two variable tangents are drawn to a conic section so that the portion of a fixed tangent, intercepted between them, subtends a right angle at a fixed point. Prove that the locus of the point of intersection of the variable tangents is a straight line.

If the fixed point be a focus, the locus will be the corresponding directrix.

5.

Chords are drawn to a conic, subtending a right angle at a fixed point; prove that they all touch a conic, of which that point is a focus.

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2

6. Three given straight lines BC, CA, AB, are intersected by two other given straight lines in A ̧, Д ̧; B, B ̧; С1, C, respectively. Prove that a conic can be described touching the six straight lines ▲Ä ÃÄ BB BB, CC, CC

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7. A, B, C, S, are four fixed points, SD is drawn perpendicular to SA, intersecting BC in D, SE perpendicular to SB, intersecting CA in E, SF perpendicular to SC, intersecting AB in F. Prove that D, E, F lie in the same straight line.

Prove also that the four conics which have S as a focus, and which touch the three sides of the several triangles ABC, AEF, BFD, CDE, have their latera-recta equal.

8. Two conics are described with a common focus and their corresponding directrices fixed; prove that, if the sum of the reciprocals of their latera-recta be constant, their common tangents will touch a conic section.

9. A conic is described touching three given straight lines BC, CA, AB, so that the pair of tangents drawn to it from a given point 0, are at right angles to each other. Prove that it will always touch another fixed straight line; and that, if this straight line cut BC, CA, AB in D, E, F respectively, each of the angles AOD, BOE, COF is a right angle.

Prove also that the polar of O with respect to this conic will always touch a conic, of which O is a focus.

F.

a

10. OA, OB, are the common tangents to two conics having common focus S, CA, CB are tangents at one of their points of intersection, BD, AE tangents intersecting CA, CB in D, E. Prove that S, D, E lie in the same straight line.

11. Any triangle is described, self-conjugate with regard to a given conic; prove that, if a conic be described, touching the sides of this triangle, and having the centre of the given conic as a focus, its axis-minor will be constant.

12. Prove that two ellipses, which have a common focus, cannot intersect in more than two points.

13. If a system of conics be described, passing through four given points, four fixed straight lines may be found, such that the chord of each, intercepted by any conic of the system, subtends a right angle at one of the points.

CHAPTER VII.

TANGENTIAL CO-ORDINATES.

1. IN the systems of co-ordinates with which we have hitherto been concerned, we have considered a point as determined, directly or indirectly, by means of its distances from three given straight lines; and we have regarded a curve as the aggregation of all points, the co-ordinates of which satisfy a certain equation. It is equally possible, however, to consider a straight line as determined by means of its distances from three points, which distances may be termed its coordinates; and to regard a curve as the envelope of all straight lines, the co-ordinates of which satisfy a certain equation.

This system is closely connected with the theory of reciprocal polars. In fact, it may be looked upon as a means of so interpreting equations as at once to obtain the results which the method of reciprocal polars would deduce from the ordinary method of interpretation. The equations are the reciprocals of those described in Chapter v. with respect to x2 + y2 + ≈2 = 0.

We may then define the co-ordinates of a straight line to be the perpendiculars let fall upon it from three given points A, B, C. The lengths of these perpendiculars we will denote by the letters p, q, r, respectively, the lengths BC, CA, AB being represented as before by the letters a, b, c, and the angles of the triangle of reference ABC being denoted by A, B, C, and its area by A.

2. Any two co-ordinates, q and r for example, will be considered to have contrary signs if the line of which they

EQUATION OF A POINT.

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are the co-ordinates cuts the line BC in a point lying between B and C, otherwise to have the same sign. Thus, the internal bisector of the angle A has its co-ordinates of contrary signs, the external bisector of the same sign. The sign of p relatively to q and r will be determined in the same

manner.

If D be any point on the line BC, q, r, the co-ordinates of any line passing through it, and BD=a1, CD=a2, distances measured along the line BC from B to C being considered positive, and from C to B negative, it will readily be seen that

Since this is a relation between the co-ordinates of any line passing through the point D, it may be considered as the equation of the point D.

If D be the middle point of BC, a, a,, hence it appears that the middle points of the sides of the triangle of reference are represented by the equations,

q+r=0, r+p=0, p+q=0.

It may also be proved that the points where the internal bisectors of the angles meet the opposite sides, are represented by

bq+cr=0, cr+ap=0, ap+bq=0.

The points where the external bisectors of the angles meet the opposite sides, by

bq-cr=0, cr-ap=0, ap-bq = 0.

The feet of the perpendiculars from the angular points on the opposite sides, by

tan B+r tan C=0, r tan C+ p tan A = 0,

p tan A+q tan B=0.

The points of contact of the inscribed' circle, by

3. We shall next prove the following proposition; that if O be any given point within the triangle ABC, then the co-ordinates p, q, r (their signs being taken in the manner already explained) of any straight line QPR, passing through it, will be connected by the following equation,

ABOC.p+ACOA.q+▲AOB. r = 0.

and the trilinear co-ordinates of O, §, n, C.

Then

: ABOC: ACOA: AAOB.

And, since lies on QPR,

l+m+ng=0.

Again, since p is the distance from the point (1, 0, 0) to the line, (1, m, n),

.: (1.2A)2={(l—m) (l—n) a2+(m—n) (m—1) b3+ (n−1)(n—m)c3} p3. Similar equations hold for Q and R, hence

ABOC.p+ACOA.q+AAOB.r= 0.

This equation may be regarded as the equation of the

point 0.