11. Def.—A pencil of rays is a system of straight lines (rays) passing through a point (the vertex).

Thus OA, OB, OC, OD, Fig. 13, form a pencil of rays.

12. Def.-A harmonic pencil is a pencil of four rays which pass through the harmonic points of a line.

13. Any transversal is cut harmonically by a harmonic pencil. Hint.—Let A, B, C, D be harmonic points, P and p the perpendiculars

The ratio of the areas of two corresponding triangles as aOc and AOC is

14. Each diagonal of a complete quadrilateral is divided harmonically by the other two.

(1)

Hint. Since BN is a transversal cutting the sides of the triangle ACF,

(2) Also

AB.CN.FEBC.NF. EA.

AB.CM.FE = BC.MF.EA.

CN NF

By dividing (1) by (2).

CM

Therefore

15. If two harmonic pencils have one pair of corresponding rays coincident, the intersections of the other three pairs of corresponding rays are in a straight line.

BD

FIG. 15

Hint.-Use the method of reductio ad absurdum.

SOME PROPERTIES OF CIRCLES

16. The product of the perpendiculars drawn from a point on a circle to two tangents is equal to the square of the perpendicular drawn from the point to their chord of contact.

Hint.-Let RT, R'T" be the tangents and TT" their chord of contact. A circle can be circumscribed about each of the quadrilaterals APRT and APR' T', since the sum of the opposite angles in each is equal to two right angles.

Hence angle ARP=ATP=PT'R'=R'AP. PAR.

Likewise angle AR'P=

Therefore the triangles ARP and R'AP are similar and AP2 = PR.PR'.

* The word circle instead of circumference is used except where ambiguity would result.

17. Def.-The angle at which two circles cut each other is the angle between the tangents drawn at the point of intersection.

B

/A

FIGS. 17 AND 18

18. Def.-If two circles cut each other at right angles they are said to cut orthogonally.

19. If the square of the distance between the centres of two circles is equal to the sum of the squares of their radii, the circles cut each other orthogonally and conversely.

20. If a circle be circumscribed about a triangle, the lines joining the extremities of the diameter which is perpendicular to the base, to the vertex, are the internal and external bisectors of the vertex angle. Hint.-Angle DCE is a right angle.

21. Defs. The point of intersection of the direct common tangents of two circles is their external centre of similitude; the point of intersection of their inverse common tangents, their internal centre of similitude.

The centres of similitude are on the line of centres of the circles, and divide that line externally and internally in the ratio of the radii.

ӨӨ

EXTERNAL CENTRE

FIG. 21

INTERNAL CENTRE

22. The six centres of similitude of three circles are three by three on four straight lines.

The three external centres of similitude are in a straight line, and each pair of internal centres of similitude is in a straight line passing through an external centre of similitude.

2.

Hint.-Let S, S2, S3 be the external, T1, T2, T3 the internal centres of similitude; also let R1, R2, R3 be the radii of the circles.

23. Cor.-If a variable circle touch two fixed circles, the chord of contact passes through an external centre of similitude of the fixed circles; for each point of contact is a centre of similitude of the variable circle and one of the fixed circles.

INVERSION

24. Def.-Two points are inverse to each other with respect to a given centre of inversion if they are in the same straight line with this centre, and if the product of their distances from it is equal to a con

stant.

Two curves are inverse to each other if the successive points of the one invert into the successive points of the other with respect to a given centre.

P

FIG. 24

Qis the inverse of P with respect to the centre, O if OP =

K 0Q

The curve Y is the inverse of the curve X, if, for every point P of X there is a point of Y such that OP.OQ=K.

25. The inverse of a circle is a straight line if the centre of inversion is on the circle.

Hint.-OP.OQ=OA.OB=K.

FIG. 25