289°. Def.—With reference to the centre O (Fig. of 285°), X and Y', as also X' and Y, are called antihomologous points. Similarly with respect to the centre O', U' and Z, as also U and Y', are antihomologous points.

Let tangents at X and Y' meet at L. Then, since CX is || to C'X', 4CXY=4C'X'Y' = 4C'Y'X'. But LXY is comp. of ¿CXY and ¿LY'X' is comp. of ¿C'Y'X'.

ALXY' is isosceles, and LX=LY'.

L is on the radical axis of S and S'.

Similarly it may be proved that pairs of tangents at Y and X', at U and Y', and at U' and Z, meet on the radical axis of S and S', and the tangent at U passes through L.

.. tangents at a pair of antihomologous points meet on the radical axis.

Cor. 1. The join of the points of contact of two equal tangents to two circles passes through a centre of similitude of the two circles.

Cor. 2. When a circle cuts two circles orthogonally, the joins of the points of intersection taken in pairs of one from each circle pass through the centres of similitude of the two circles.

But OX'. OY'=the square of the tangent from O to the circle S' and is therefore constant.

OX.OY' = OT2 a constant.

.. X and Y' are inverse points with respect to a circle whose centre is at O and whose radius is OT'

√

Def. This circle is called the circle of antisimilitude, and will be contracted to O of ans.

Evidently the circles S and S' are inverse to one another with respect to their of ans.

For the centre O' the product O'U.O'Y' is negative, and the O of ans. corresponding to this centre is imaginary.

Cor. 1. Denoting the distance CC' by d, and the difference between the radii (r' − r) by d, we have

1. When either circle becomes a point their of ans. becomes a point.

2. When the circles S and S' are equal, the O of ans. becomes the radical axis of the two circles.

3. When one circle touches the other internally the of ans. becomes a point-circle. (d=d.)

4. When one circle includes the other without contact the O of ans. is imaginary. (d<d.)

Cor. 2. Two circles and their circle of antisimilitude are co-axal.

(263°)

Cor. 3. If two circles be inverted with respect to their circle of antisimilitude, they exchange places, and their radical axis being a line circle co-axal with the two circles. becomes a circle through O co-axal with the two.

The only circle satisfying this condition is the circle of similitude of the two circles. Therefore the radical axis inverts into the circle of similitude, and the circle of similitude into the radical axis.

Hence every line through O cuts the radical axis and the circle of similitude of two circles at the same angle.

291°. Def.-When a circle touches two others so as to exclude both or to include both, it is said to touch them similarly, or to have contacts of like kind with the two. When it includes the one and excludes the other, it is said to touch them dissimilarly, or to have contacts of unlike kinds with the two.

292°. Theorem.--When a circle touches two other circles, its chord of contact passes through their external centre of similitude when the contacts are of like kind, and through their internal centre of similitude when the contacts are of unlike kinds.

Proof. Let circle Z touch circles S and S' at Y and X'. Then CYD and C'X'D are lines.

(113°, Cor. 1)

Let XYX'Y' be the secant through Y and X'. Then

4CXY=4CYX=¿DYX'=¿DX'Y=4CX'Y'.

.. CX and C'X' are parallel, and X'X passes through the external centre of similitude O.

(285°) Similarly, if Z' includes both S and S', it may be proved that its chord of contact passes through O.

Again, let the circle W, with centre E, touch S' at Y' and S at U so as to include S' and exclude S, and let UY' be the chord of contact. Then

4CVU=4CUV=LEUY'=LEY'U,

.. EY' and CV are parallel and VY' connects them transversely; .. VY' passes through O'.

q.e.d.

Cor. 1. Every circle which touches S and S' similarly is cut orthogonally by the external circle of antisimilitude of S and S'.

Cor. 2. If two circles touch S and S' externally their points of contact are concyclic. (116°, Ex. 2) But the points of contact of either circle with S and S' are antihomologous points to the centre O.

.'. if a circle cuts two others in a pair of antihomologous points it cuts them in a second pair of antihomologous points.

Cor. 3. If two circles touch two other circles similarly, the radical axis of either pair passes through a centre of similitude of the other pair.

For, if Z and Z' be two circles touching S and S' externally, the external circle of antisimilitude of S and S' cuts Z and Z' orthogonally (Cor. 1) and therefore has its centre on the radical axis of Z and Z'.

Cor. 4. If any number of circles touch S and S' similarly, they are all cut orthogonally by the external circle of antisimilitude of S and S', and all their chords of contact and all their chords of intersection with one another are concurrent at the external centre of antisimilitude of S and S'.

293°. Theorem.-If the circle Z touches the circles S and S', the chord of contact of Z and the radical axis of S and S' are conjugate lines with respect to the circle Z.

Proof. Let Z touch S and S′ in Y and X' respectively. The tangents at Y and X' meet at a point P on the radical axis of S and S'. (178°)

But P is the pole of the chord of contact YX'.

.. the radical axis passes through the pole of the chord of contact, and reciprocally the chord of contact passes through the pole of the radical axis (267°, Def.) and the lines are conjugate.

q.e.d.

AXES OF SIMILITUDE.

294°. Let S1, S2, S3 denote three circles having their centres A, B, C and radii r1, 72, 73, and let X, X', Y, Y', Z, Z' be their six centres of similitude.

Similarly it is proved that the triads of points XY'Z', YZ'X', ZX'Y' are collinear.

Def.-These lines of collinearity of the centres of similitude of the three circles taken in pairs are the axes of similitude of the circles. The line XYZ is the external axis, as being external to all the circles, and the other three, passing between the circles, are internal axes.

Cor. 1. If an axis of similitude touches any one of the circles it touches all three of them. (286°)

Cor. 2. If an axis of similitude cuts any one of the circles it cuts all three at the same angle, and the intercepted chords are proportional to the corresponding radii.

Cor. 3. Since XYX'Y' is a quadrangle whereof XX', YY', and ZZ' are the three diagonals, the circles on XX', YY', and ZZ' as diameters are co-axal. (277°, Cor. 2)

the circles of similitude of three circles taken in pairs. are co-axal.