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Scholium.-If any point not in AB be taken, and a pencil of four rays drawn through A, C, B, D, it is said to be a harmonic pencil.

Proposition 19.

Problem. To find the locus of all points whose distances from two given points are in a given ratio.

Let A and B be the points, and let M: N be the given ratio.

Divide (Prop. 18) AB harmonically in C and D; on CD describe a circle. The circumference of this circle will be the required locus.

Take any point, P, and

F

D

B E

join PA, PC, PB, PE, and PD. Because AB is divided harmonically in C and D,

or,

AD AC:: BD: BC,

AE+EC: AE-EC:: CE+EB: CE- EB;

hence, (IV. 11), AE: EC:: EC: EB,

or,

AE: EP :: EP : EB.

Therefore the sides about the common angle E of the triangles AEP, BEP are proportional; hence the triangles are similar, and the angle BPE is equal to A.

Also

But

ECP= CAP+ CPA, and its equal

EPC-EPB+BPC.

EPB-CAP; ... APC-BPC,

and (V. 3), AP : PB :: AC: CB :: M : N.

But P is any point of the circumference; hence the circumference is the required locus.

Corollary 1.-Because CPD is a right angle, and PC bisects the angle APB, PD must also bisect the angle BPF.

Corollary 2.-If the angles APB, BPF be bisected, then AB is divided harmonically in C and D; and, conversely, if AB be divided harmonically in C and D, the angles APB, BPF will be bisected by PC and PD.

SECTION VI.

POLE AND POLAR IN THE CIRCLE.

DEFINITIONS.

IF from a point, O, without the circle ABD, the secant OAB

be drawn, and C, the harmonic conju

gate of O with respect to A and B, be

taken, then the locus of C, as OAB revolves about O, is called the polar of the point ; and O is the pole of this locus.

B

Proposition 20.

Theorem.-The polar of a point is a straight line at right angles to the diameter through the point.

Let O and C be harmonic conjugates with respect to A and B on the circumference ABD. Draw CE at right angles to the diameter DE; and on OC describe a semicircle passing

through F. Join BF, and let it cut the circle in G; join

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FBH, and therefore OF

is divided (Prop. 19,

Cor. 2) harmonically in D and E, and F is the harmonic conjugate of 0.

Hence F is a fixed point, and C is always in the perpendicular to OE through F; therefore CF is the polar of the pole O.

Corollary 1.-The radius of the circle is a mean proportional between the distances of the centre from the pole and the polar.

For, because

as in last Prop.,

or,

OE OD: FE: FD,

OK: DK:: DK: FK,

OK. FK DK2 = R2.

==

Corollary 2.-The pole and polar are on the same side of the centre; if the pole be exterior, the polar is interior to the circle, and vice versa.

Corollary 3.-The polar passes through the point of contact of the tangent through the pole.

Corollary 4.-The polar of the centre is at infinity, and the polar of a point at an infinite distance passes through the centre. Corollary 5.-The polar of a point on the circle is a tangent through this point.

Proposition 21.

Theorem.-The polars of all points of a straight line pass through the pole of that line; and the poles of all straight lines which pass through a fixed point are in the polar of that point.

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CD. From the similar triangles DCE, FCG,

=

CE. CGCD. CF.

G

Hence CD. CF CH2 and FG is (Prop. 20, Cor. 1) the polar of D.

Hence the polar of any point, D, of the line AB passes through G, and the pole, G, of any line, AB, which passes through D is in the polar FG of the point D.

Scholium. From this it results that if the angles of the polygon ABCDE be the poles

of the sides of the polygon MNOPQ, the angles of the second polygon are the poles of the sides of the first. Each polygon is said to be the reciprocal polar of the other with respect to the circle, which is called the directing circle.

M

If one polygon be inscribed in the circle, the other will be circumscribed about it, the sides of the latter passing through the angles of the former.

The angles of either may be determined from the sides of the other, or the sides of either from the angles of the other.

A double set of theorems may be deduced, in which the properties relating to the lines of one may be converted into properties relating to the points of the other, and vice versa. These theorems belong principally to curves of which Elementary Geometry does not treat, and will not be considered here.

SECTION VII.

RADICAL AXES.

DEFINITIONS.

1. THE power of a point with reference to a circle is the

rectangle of the segments of a secant through the point, formed by the circumference of the circle.

Thus the power of P with reference to ABCD is PA. PB; of P', is PC. P'D.

2. Two circles cut each other orthogonally when the radii to the point of intersection are at right angles to each other.

That is, when ACB is a right

angle.

B

D

P

A

3. The radical axis of two circles is the locus of all points whose powers, with reference to the circles, are equal.

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