from which the position of the centre and magnitude of the radius of the inverse circle may be determined.

COR. If the centre of inversion is on the circle; d=r and r', thus verifying that the inverse of a circle. from any origin on its circumference is a right line.

124. Problem.-To invert two circles such that the ratio of the radii of their inverses may be a given quantity K.

Let r1, r2 be the radii of the given circles; d1, d1⁄2 the distances of their centres from the origin 0; R the radius of inversion; t1, t2 the tangents, real or imaginary, from 0 to the given circles. Then if p1, p2 denote the radii of the inverse circles, we have, by Art. 123,

P1= R2r/t2 and p2- R2r/t22.

Dividing these equations,

=

The centre of inversion is therefore on a locus such that tangents drawn from any point on it to the given circles have a constant ratio; i.e. a circle coaxal with them.

COR. Any two circles may be inverted into equal circles; and the locus of the centre of inversion is either circle of antisimilitude.

For when p1=p2; t2/t2=r/r2; therefore, etc. (Art. 114, 2°.)

Otherwise thus:-Since a circle and two inverse figures invert into a circle and two inverse figures; if the origin

be taken on either circle of antisimilitude this circle inverts into a line. Therefore any two figures the inverse of each other with respect to a circle invert into reflexions of each other with respect to a line. (Art. 121, Ex. 6.)

EXAMPLES.

1. Show how to invert any three circles into equal circles. [The centres of inversion are the points of section of the circles of antisimilitude of the given ones taken in pairs.]

2. How many centres of inversion are there in the solution of Ex. 1?

[The three external circles of antisimilitude are coaxal (Art. 88, Ex. 13), and therefore meet in two real or imaginary points. Also since every two internal and one external circles of antisimilitude are coaxal, there are in all eight centres of inversion real or imaginary.]

3. Any three circles are unaltered by inversion with respect to their common orthogonal circle. For this reason the latter has been named the Circle of Self-Inversion of the given ones.

4. To invert the sides of a triangle into

a. Three equal circles.

B. Three circles whose radii have any given ratios p: q: r. [a. The centres of the in- and ex-circles are the four origins. B. The distances of the origin from the sides are in the inverse ratios p: q: r.]

125. Theorem.-The tangents at corresponding points A and A' of two inverse figures make equal angles with their line of connexion AA'.

For take the corresponding points B and B' on the curves which are consecutive to A and A'. Join AA' and BB'; they each pass through 0.

The lines AB and A'B' joining consecutive points may be regarded as tangents to the respective curves; also

since ABA'B' is a cyclic quadrilateral and the angle at O indefinitely small, we have (Euc. III. 22)

B40=0B' Α' = ΑΑ' Β',

therefore TAA' is an isosceles triangle.

126. Theorem. The angle of intersection of two curves is similar to that of their inverses at the corresponding point.

For the angle between any two curves is the angle between the tangents at their points of intersection.

But the tangents determine two isosceles triangles (Art. 125) on the line AA'; therefore, etc.

If the centre of inversion is external or internal to both circles the angle remains unaltered; if on the other hand it is external to either and internal to the other, the angles of intersection before and after inversion are supplemental.

*The angle of intersection of two circles undergoes as a figure no change of form under the process of inversion, but often does as a magnitude, change into its supplement, under that process.

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'In the application of the theory of inversion to the geometry of the circle, this circumstance must always be attended to.

"The two cases of contact, external and internal, come of course under it as particular cases; and in but one case alone, that of orthogonal intersection, which presents no ambiguity, can the precaution ever be entirely dispensed with." Townsend's Modern Geometry of the Point, Line, and Circle, Art. 407.

127. Amongst the various results which follow from the preceding Articles, we note

1°. Any two circles meeting at an angle a invert from either point of intersection into two lines inclined

at the same angle, e.g. two orthogonal circles into two lines at right angles.

2o. Three mutually orthogonal circles, e.g. the three real polar circles of the triangles formed from an orthocentric system of points, invert from any of their points of intersection into a circle and two perpendicular diameters.

3o. Any three circles invert from any centre on their

common orthogonal circle into three others whose centres are collinear; the line of collinearity being the inverse of the common orthogonal circle. 4°. A system of circles having more than one orthogonal circle inverts into a system having more than one orthogonal line.

5°. In 4° the intersections of the common orthogonal circles are evidently the limiting points of the given system which is coaxal. (Art. 86.)

Hence for any centre of inversion:

a. A coaxal system inverts into a coaxal system; or b. A circle and a pair of inverse points invert into a circle and a pair of inverse points;

and for a centre of inversion at either of the limiting points:

c. A coaxal system inverts into a concentric system,

the common centre being the inverse of the second limiting point with respect to the circle of inversion.

6°. A system of concurrent lines inverts into a coaxal system of the common point species, the common

points being the centre of inversion and the inverse of the point of concurrence.

7°. An angle and its bisectors invert into two circles and their circles of antisimilitude. (Art. 109.)

2

8°. If two circles, concentric with the extremities of the third diagonal of a cyclic quadrilateral, are described cutting the given one orthogonally; they are mutually orthogonal, and their points of intersection 01 and О, are therefore inverse points with respect to the given circle. Hence Hence if we take 01 and O2 as centres of inversion we arrive at the following results:-The three circles invert into a circle and two rectangular diameters; the vertices of the quadrilateral, which are inverse points with respect to the circles, invert into inverse points in the same order with respect to the lines, i.e. form the vertices of a rectangle. Thus the vertices of any cyclic quadrilateral may be inverted into those of a rectangle, and the centres of inversion are inverse points with respect to the circle.

9°. A circle may invert into a circle having its centre at a given point A.

For let A' the inverse of A be the centre, and AA' the radius of inversion. Then the given circle and pair of points A and A' inverse to it, invert into a circle and a pair of inverse points; but the inverse of the centre of inversion A' is at infinity; therefore A is the centre of the inverse circle.