THEOREM (5)—If a variable circle touches two fixed circles, the join of the points of contact goes through the external centre of similitude of the fixed circles, when the contacts are of the same kind; and through the internal centre, when of different kinds.

Let X be variable, and touch fixed Os A, B, in Q, p respectively.
Let pQ cut BA in S, and O A again in P.

Cor. The tangent to OX from S is constant. For the sq. on it which is const. by Theorem (4).

THEOREM (6)—The six centres of similitude, which are got by taking three circles in pairs, are so situated that—

(a) the joins of the centre of each circle with the internal centre of similitude of the other two are concurrent:

(B) the external centre of similitude of any pair, and the two internal centres of similitude of the other two pairs, are collinear:

(y) the three external centres of similitude are collinear.

Let A, B, C be centres of three Os;

the respective ext. and int. centres simil. of O' B, C ;

Def. The line S, S, S, is called the external axis of similitude of the three circles; and the lines S, 0, 0, S2 σ1 σ, S, σ, σ2 are called the three internal axes of similitude.

EXERCISES ON CENTRES OF SIMILITUDE.

1. The centroid and orthocentre of a triangle are respectively the internal and external centres of similitude of its circum-circle and N. P. circle.

2. D, E, F are the points of contact of the in-circle, respectively opposite the corners A, B, C of a triangle; if X is taken in CB, so that CX, BD are equal; and if AX cuts the in-circle in P, Q (of which P is nearer to A), then

3. X, Y are the respective points of contact with BC, of the in-circle, and an ex-circle, of a triangle ABC; if YP is perpendicular to AX, then P is on the circumference of the ex-circle.

4. Through the external centre of similitude of two circles A, B, a variable line is drawn, meeting the circles in P, Q, p, q; if a circle is drawn touching A, B at non-corresponding points P, q; and another circle touching them at p, Q; then the difference of the radii of these last two circles is equal to the sum of the radii of A, B.

5. Given two non-intersecting circles; show that of all lines, parallel to a given direction, which meet the circles, the one through the internal centre of similitude has one of its segments, intercepted between the two circumferences, maximum, and one minimum.

NOTE-Use Theorem (3) Cor.

Def. The circle on So [fig. of Theorem (1)] as diameter, is called the circle of similitude of circles A and B.

6. Show that the circle of similitude of two circles is the Locus of points at which the circles subtend equal angles.

NOTE-By reference to vi. Addenda (15) it will be seen that the circle of similitude is such a Locus as is there investigated; and that if X is any point on its circumference, XA: XB radius of A: radius of B.

=

7. If PX, PY are tangents to two circles from any point P on their circle of similitude; and if XY meet the circles again in x, y; then will X X and Y y be equal. (Chasles, Géométrie Supérieure, p. 525.)

8. In Theorem (5) all the variable circles are cut orthogonally by a fixed circle.

9. Two fixed circles are each touched by two variable circles; if the variable circles also touch each other, find the Locus of their point of contact.

NOTE-Use Theorem (5).

10. If Tt is a common tangent, and PQpq a common line of section of two circles, drawn through the same centre of similitude, and so that P, p and Q, q are corresponding points; then

II. If from S, a centre of similitude of two circles, two lines SPQpq, SXY xy, are drawn to cut the circles; so that P, Q, X, Y are on one circle; and p, q, x, y are the corresponding points on the other; and if each pair of points on the same circle are joined, then

1o, the join of any pair, as PX, is parallel to the join px of the corresponding pair :

2o, the quadrilaterals PYxq, QXyp, QYxp, PXyq are cyclic :

3o, the join of any pair of points, as PX, meets the join qy of the noncorresponding pair at a point such that the tangents from it to the circles are equal. [Cf. Section iv. (1) for the Locus of these points.]

12. If X is the centre of the circle of similitude of two circles A, B, whose respective radii are a, b, show that

Hence deduce that, if C is a third circle; and Y, Z the centres of the circles of similitude of B, C and of C, A; then X, Y, Z are collinear.

Def. The two circles round the two centres of similitude of a pair of circles as centres, the squares on whose radii are equal to the corresponding rectangles of anti-similitude, are called the circles of anti-similitude.

13. Every circle orthogonal to two circles is orthogonal at once to their circle of similitude, and their two circles of anti-similitude.

14. If A, B, C are any three circles; X a circle which touches them all internally, and Y a circle which touches them all externally; prove that— 1o, the radical axis (see Section iv.) of X, Y, is the axis of external similitude of A, B, C ; and

2o, the internal centre of similitude of X, Y, is the radical centre of A, B, C. How should the Theorems be modified, when the contacts are not all of the same kind?

15. If a variable circle cuts two fixed circles, at equal angles, the join of a pair of non-corresponding points of intersection goes through the external centre of similitude of the fixed circles.

SECTION iv-CO-AXAL CIRCLES.

Def. That line which is perpendicular to the line of centres of two circles, and divides the distance between their centres (internally or externally) into segments, the difference of the squares on which is equal to the difference of the squares on the radii, is called the radical axis of the circles.

THEOREM (1) The radical axis of two circles is the Locus of points from which tangents to the circles are equal.

(1)

(2)

R

A

CB

Take any two Os C, B-the radius of OC being the greater-and let A be the pt. in CB, fig. (1), or CB produced, fig. (2), for which

Cor. (1). When two Os intersect, their radical axis is also their common chord.