Imágenes de páginas
PDF
EPUB

gonal circle of the three cuts them at once at equal and supplemental angles, it belongs to each of the four coaxal systems.

5. If two circles A and B touch with similar contacts three others X, Y, Z, the radical axis of 1 and B is the line joining the ex-centres of similitude of X, Y, Z taken in pairs.

[A particular case of the foregoing.]

6. The eight circles that can be described to touch three given ones arrange themselves in pairs coaxal with the four axes of similitude of the given ones.

7. In Ex. 5 the chords of the three circles joining the points of contact with the two meet at the in-centre of similitude of A and B, and therefore at the radical centre of X, Y, Z.

8. The chords of contact pass through the poles of the radical axis of A and B with respect to each of the circles X, Y, Z.

[For the tangents at the extremities of the chord of contact of I being equal intersect on the radical axis of A and B.]

[graphic][subsumed][subsumed][subsumed][subsumed][subsumed][subsumed]

NOTE.-Gergonne deduces by means of the foregoing properties a simple geometrical construction for the eight circles of contact of

three given ones X, Y, Z. The circles having similar contacts are found as follows:-Find the ex-centres of similitude of X, Y, Z taken in pairs; the line I joining them is the radical axis of the required circles A and B. Next find C' the centre of the common orthogonal circle of the given ones. C'' is the in-centre of similitude of A and B. Now obtain the inverses X', Y', Z' of L with respect to X, Y, Z respectively. Join C'X', C'Y', and C'Z'; these lines meet the given circles at the required points of contact; therefore, etc. The remaining circles may be similarly found.

Otherwise, thus :-) By Casey's relation in Art. 7, if we number the given circles 1, 2, 3 and let 4 be the required point of contact with 1, we have the ratio of the tangents from 4 to 2 and 3, a given quantity k. Similarly for the second circle which has the similar contacts with the three given ones, the ratio of the tangents from its point of contact (5) to 2 and 3-the same ratio k; therefore, etc. (Art. 88, Cor. 1).

9. Let A1A2, B1B2 be the extremities of the common diameter of two circles; M, N their limiting points; prove that the circles on A1B1, A2B, MN as diameters are coaxal.

[For their centres are collinear, and they each cut the internal circle of antisimilitude orthogonally (Art. 114, 4°); therefore, etc.]

10. A variable circle cutting three given ones at equal angles passes through two fixed points, real or imaginary.

[For it cuts the external circles of antisimilitude of the given ones taken two and two orthogonally, and these (Art. 88, Ex. 13. 2°) are coaxal; therefore the variable circle passes through their limiting points, real or imaginary.]

11. Two variable circles X and Y touch externally two fixed circles A, 1 and B, r2 at four points B1, A2 and A, B1 in a right line; prove that

ao. The line joining their centres passes through a fixed point.

B. The sum of their radii is constant.

7. The foot of their radical axis describes a circle.

[a. Since the diagonals of a parallelogram bisect each other, XP bisects and is bisected at the middle point Z of A B.

[graphic][subsumed][subsumed][ocr errors][subsumed][subsumed][ocr errors][subsumed][subsumed][subsumed][subsumed][ocr errors]

B. Let L be the radical axis of A, 71, and B, 2; then XL/p= YL/pi

=const. (Art. 88, Ex. 7), and therefore

XL+ YL p+pi

=

= const., but the

numerator is constant by a° (=2ZL); therefore, etc.

7°. The circle on CZ is evidently the locus.]

12. Circles are described touching two fixed circles (as in Fig. of Ex. 8); find the locus of the limiting points of these circles taken in pairs.

[The internal circle of antisimilitude of the two given circles (Art. 114, 3°).]

12a. Circles are described touching one another, and each touching two given circles; find the locus of their points of contact.

[The points of contact are the coincident limiting points of the touching circles; hence the required locus is the internal circle of antisimilitude of the two given ones.]

13. If n points be taken on a circle, prove that (1) the mean centres of the n systems of n-1 points formed by omitting each

point in succession, lie on a circle S,; (2) if another point be taken on the original circle, the centres of the n+1 circles (Sn) obtained by omitting each point in succession lie on an equal circle; and so on ad infinitum. (St. Clair.)*

[Let G be the mean centre of the system of n points.

Produce AG to a, making AG: Ga=n-1: 1; then a is the mean centre of the n 1 points formed by excluding A. In the same manner we get BG: Gb-n-1:1, etc.; hence the points a, b, ... lie on a circle; and G is a centre of similitude of the locus circle and the given one.

*Educational Times, February, 1891.

CHAPTER XI.

INVERSION.

SECTION I

INTRODUCTORY.

115. It has been seen (Art. 74) that the inverse of every point on a line with respect to a circle lies on a circle described on the line joining the centre of the given circle with the pole of the line.

This circle is said to be the inverse of the line with respect to the given circle; and it may be generally inferred that the inverse of a line is a circle passing through the centre of the given circle; and conversely. This latter is named the Circle of Inversion, and its centre the Origin or Centre of Inversion.

We shall now proceed to discuss the inversion of a system of points which are not collinear. Take the simplest case the vertices of a triangle ABC. Let their inverses with respect to a circle of inversion O, r be respectively A', B', C'.

It is obvious that the three quadrilaterals BCB'C', CAC'A', ABA'B' are cyclic; hence we have the angular relations :—

A'C'O=OAC, B'C'O=OBC, etc. (Euc. III. 22),

« AnteriorContinuar »