It follows, because their centres lie on a line, that they pass

through a second point, the reflexion of the first with respect to the line, i.e., they are coaxal.

18. Having given the base and ratio of sides of a triangle, the locus of the vertex is a circle to which the extremities of the base are inverse points (origin at either).

The line joining the centre of a conic to the foot of the perpendicular from focus on any tangent is constant.

The locus of the foot of the perpendicular is called the Auxiliary Circle of the conic. The circle and conic evidently touch at the extremities of the major axis.

Since the centre of a parabola is at infinity, its auxiliary circle degenerates into the tangent at the vertex.

19. Common tangents to two circles subtend right angles at either common inverse point.

20. The feet of the perpendiculars from any point on a circle on the sides of an inscribed triangle are collinear.

The perpendiculars through the vertices of a triangle, escribed to a parabola, to the lines joining them to the focus are concurrent;

in other words, the circum-circle of a triangle described about a parabola passes through the focus (cf. Ex. 18). We infer that the circum-circles of the four triangles formed by four tangents (that is any four lines whatever) meet in a point.

It follows also, since any point (origin) on the circum-circle and the orthocentre are equidistant from the Simson line of the point, that the locus of the orthocentre of a variable triangle escribed to a parabola is the directrix

21. Having given base and vertical angle, the locus of the vertex of the triangle is a circle. (Euc. III. 21.)

If the extremities of a variable line, which subtends a constant angle at a fixed point, move on two fixed lines, it envelopes a Iconic to which these lines are tangents. It therefore cuts them equianharmonically. 22. Since inverse segments subtend similar angles at any point on the circle, the segments of a line drawn across two circles subtend similar angles at either common inverse point.

The pairs of tangents to confocal conics from any point are equally inclined.

Confocal conics have pairs of imaginary common tangents passing through the foci.

The poles of a line with respect to a system of confocal conics are collinear.

The locus of the poles is a line perpendicular to the given one.

The sum of the squares of the reciprocals of the distances of the foci from two rectangular tangents is constant ;

hence if P1, P2, T1, T2 denote the distances of the foci from the

2.

tangents 21/p2 = constant.

27. In Ex. 26, if the square of the radius of reciprocation is the power of the point with respect to the circle.

P12+p22+π12+π22=constant ;

or the locus of the intersection of rectangular tangents is a concentric circle (Director Circle).

28. From the properties of the conic, rectangular tangents, director circle, centre and line at infinity.

A variable chord of a conic which subtends a right angle at any point envelopes a conic; and the focus and directrix of the envelope are pole and polar with respect to the given conic.

If the point is on the given conic the envelope reduces to a point * on the perpendicular to the tangent passing through its point of contact. (The Normal.)

29. The base BC of a triangle ABC inscribed in a circle is fixed and the origin taken at its pole. Applying the formula of Art. 79, Ex. 10, we have the area of the reciprocal triangle constant, hence :the area cut off by any tangent with the asymptotes is constant. And conversely, given the vertical angle in position and area of a triangle, the envelope of the base is a conic; and the sides are divided equianharmonically by the extremities of the base.

30, Show by reciprocating from a vertex of a self conjugate triangle with respect to a circle that

a. The sum of the squares of any two conjugate diameters of an ellipse is constant.

B. The difference of the squares of any two conjugate diameters of a hyperbola is constant.

31. Find by the methods of Art. 79, Exs. 3 and 4, the tangential equations of a conic circumscribed or inscribed to the triangle of reference.

*This is proved independently as follows: If two right lines are drawn at right angles through a fixed point and intercept a variable segment AB on a fixed tangent to a circle; the locus of the intersection of tangents through A and B is a line.

For it is a locus that can only meet the given tangent in one point; therefore, etc., by reciprocation.

CHAPTER VIII.

SECTION I.

COAXAL CIRCLES.

82. Definitions.-The Radical Axis L of two circles A, r1 and B, î2 is the line perpendicular to AB and dividing it so that AL2 ~ BL2=r ̧2~ r2. Cf. Art. 72, Ex. 3.

It follows from the definition that L is the common chord of the circles when they intersect, and we may generalize this statement by regarding the radical axis as their chord of intersection real or imaginary.

Thus all circles having a common radical axis pass through two real or two imaginary points.

Such a group is termed a Coaxal System.

83. It has been seen, Art. 72, Ex. 3, that a variable circle cutting two given ones orthogonally passes through two fixed points, viz., their common pair of inverse points; this orthogonal system is therefore coaxal; and from their mutual relations the two groups are said to be Conjugate Coaxal Systems. It is obvious that if either set possesses real points of intersection, the other does not; also the common points of one set are the common pair of inverse points with respect to the other. Art. 72, Ex. 1.

Since the line of centres AB bisects the common chord

MN it is the axis of reflexion of each common point with respect to the other.

NOTE--If two circles are concentric their radical axis is the line at infinity; therefore a system of concentric circles passes through two imaginary points at infinity.

These are called the Circular Points.

If the circles touch, their radical axis is the common tangent at the point of contact.

If the circles reduce to points, the radical axis of two points is their axis of reflexion.

; B, ra; C, rs... denote the circles of a

r;

84. Let A, coaxal system. Then, since