6. Six circles pass through two points P and Q on the circumcircle of a triangle ABC and touch the sides; prove that the points of contact X, X' ; Y, Y' ; Z, Z' lie in threes on four lines.

[Let the line joining the points P and Q cut the sides of the triangle in L, M, and N respectively, and we have obviously LX LX' and LB. LC=LX2=LX', with similar relations on the remaining sides of the triangle; therefore, etc.]

7. From any point on a given line tangents are drawn to a circle; a circle is described touching the fixed circle and variable pair of tangents to it; prove that the envelope of the polar of its centre is a circle.

8. The circle of similitude of the circum- and nine-points-circle of a triangle is that described on the interval between the centroid and orthocentre as diameter.

[Let O be the circum-centre, H orthocentre, N the nine-points centre, and E the centroid. By a well-known property of these four collinear points OE|NE=OH|NH=2=ratio of radii of circum- and nine-points-circles; therefore, etc.

[It is called the Orthocentroidal Circle of the triangle.]

MISCELLANEOUS EXAMPLES.

1. Prove that the equation of the two circles touching three given ones with contacts of similar species are

where S1, S2, S3 denote the powers of any point on either of the tangential circles with respect to the given ones.

2. If a variable chord AB of a circle is such that the sum of the tangents from A and B to another given circle is proportional to the length of AB, it envelopes a circle coaxal with the two.

3. If a variable circle touches two fixed circles and cuts a circle concentric with either in the points A and B : required to find the envelope of AB. (Dublin Univ. Exam. Papers, 1891.)

[Applying Casey's relation between the common tangents to four

circles to the points A and B and the two given circles, it follows by Ex. 2 that the envelope of AB is a coaxal circle.]

4. Prove that the circles cutting three given ones orthogonally passing through their circles, and bisecting the circumferences are coaxal.

5. Reciprocate the following theorem from a limiting point :The square of the distance of any point on a circle from a limiting point varies as its distance from the radical axis.

[The rectangle under the distances of the foci from any tangent to a conic is constant.]

6. Prove that the limiting points of any two circles lie on a pair of opposite connections of their common escribed quadrilateral.

7. If 8 denote the distance between the limiting points and y the length of their imaginary common chord, prove that d=iy.

8. If two circles whose radii are r1 and r2 are so related that a hexagon can be inscribed to one and circumscribed to the other, then

9. If an octagon can be inscribed to one and circumscribed to the other,

10. The mean centre of the vertices of a cyclic quadrilateral lies on the circumference of the nine-points-circle of the harmonic triangle of the quadrilateral. (Russell.)

11. If a variable polygon is inscribed to one circle and escribed to another, the locus of the mean centre of any number (r) of consecutive points of contact is a circle. (Weill). Cf. Art. 53, Ex. 12.

12. Prove the following extension of Weill's theorems :-If a variable polygon of any order be inscribed in a circle of a coaxal

system having all its sides touching respectively fixed circles of the system; there exists a set of multiples for which the mean centre of the points of contact of the sides with the circles is a fixed point. [Let any circle of the system be denoted by (0, r, 8) where 8 is the distance of its centre from the circumcentre of the polygon, and let a, ß, and c be the displacements of the points of contact of the sides AB, BC, CD, etc. for consecutive positions. Then, by Art. 53, Ex. 12, we have

لا

hence the mean centre of the points of contact remains fixed for the system of multiples √81/r1, √2/r2, √83/r3, etc.]

12a. The locus of the mean centre of r consecutive points of contact for their respective multiples is a circle.

[For, join the extremities of the r sides thus forming a polygon of r+1 sides, and let the last side touch a fixed circle (Or+1, Tr+1, Sr+1) of the system. (Art. 89, Ex. 4.) By Ex. 12, the mean centre of the r+1 points of contact for the corresponding multiples is a fixed point (X). Let Y be the mean centre for the r points and Z the point of contact of the last side. Then I divides the line XZ in a constant ratio, and since Z describes a circle, therefore, etc.] *

*The following is an independent proof of the generalization of Weill's theorem.

Let ABCD.. and A'B'C'D'... be any two positions of the variable polygon; T1, T2, T3, T1, T2, T3'... points of contact of the sides AB, BC, ; A'B', B'C', with the corresponding circles O1, 71, 81 ; of the system; R the point of intersection of AB and

A'B' and

...

...

the angle between them; S the intersection of AA' and BB', and the angle between them. Then AA', BB', CC... touch a circle (, p, λ) coaxal with the given system. Let L, M, N ... be its points of contact with AA', BB', CC', etc.

therefore

=

T1Tir, sin 30 _ ~1 BM
LM psin

No1. T1T' | LM =

~

and we have

PBT

ο δι

TTMN= etc.

13. If the diagonals of a cyclic quadrilateral are conjugate lines and a homothetic quadrilateral be described with their intersection as homothetic centre; prove that the consecutive pairs of sides of the one quadrilateral intersect the corresponding pairs of the other in eight points which lie on a circle coaxal with the circum-circles of the quadrilaterals. See Art. 96.

[Use the theorem of Art. 92, Ex. 2.]

i.e., multiples No1/T1, No2/T2, Nos/rs of the displacements TT, T2T2 are proportional to the sides of the polygon; therefore, etc. Bowesman.]

...

CHAPTER IX.

SECTION I.

Two SIMILAR FIGURES.

96. Two figures similar and similarly placed are said to be Homothetic, and their homologous parts are called Corresponding Points, Lines, etc. It is plain, if a line of either figure is displaced through an angle 0, that every line of it is displaced through the same angle. For let AB be displaced to A'B'. It follows (Euc. III. 21, 22), since B=B', that the angle between BC and B'C' is equal to 0.

Also, since corresponding lines meet at equal angles, a variable pair of corresponding lines passing through a pair of corresponding points A and A' intersect on the circumference of a circle described on AA' containing an angle ; and conversely.