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Cor. 1. (a) Let CD become a diameter and be I to AB.

Then AP. PB becomes AP, (96°, Cor. 5) B ..

AP2=CP.PD, and denoting AP by C, CP by v, and the radius of the circle by r, this becomes

C=v(2r-v), which is a relation between a chord of a O,

the radius of the O, and the distance CP, commonly called the versed sine, of the arc AB. (6) When the point of intersection P passes without the O

we have still, by the principle of con-
tinuity, AP.PB=CP.PD. But the Os
being now both negative we make them
both positive by writing

PA. PB=PC.PD.
Cor. 2. When the secant PAB be-
comes the tangent PT (109°), A and
B coincide at T, and PA. PB becomes
PT, .. PT2=PC.PD,

i.e., if a tangent and a secant be drawn from the same point to a circle, the square on the tangent is equal to the rectangle on the segments of the secant between the point and the circle.

Cor. 3. Conversely, if T is on the circle and PT2=PC.PD, PT is a tangent and T is the point of contact.

For, if the line PT is not a tangent it must cut the circle in some second point T' (94°). Then

PT.PT'=PC.PD=PT2. Therefore PT=PT', which is not true unless T and T' coincide. Hence PT is a tangent and T is the point of contact.

Cor. 4. Let one of the secants become a centre-line as PEF. Denote PT by t, PE by h, and the radius of the circle by r. Then

PT2=PE.PF becomes

=h(2r+h).

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EXERCISES.

1. The shortest segment from a point to a circle is a portion

of the centre-line through the point. 2. The longest segment from a point to a circle is a portion

of the centre-line through the point. 3. If two chords of a circle are perpendicular to one another

the sum of the squares on the segments between the point of intersection and the circle is equal to the

square on the diameter. 4. The span of a circular arch is 120 feet and it rises 15

feet in the middle. With what radius is it con

structed? 5. A conical glass is b inches deep and a inches across the

mouth. A sphere of radius r is dropped into it. How far is the centre of the sphere from the bottom of the

glass? 6. The earth's diameter being assumed at 7,960 miles, how

far over its surface can a person see from the top of a

mountain 3 miles high? 7. How much does the surface of still water fall away from

the level in one mile ? 8. Two circles whose radii are 10 and 6 have their centres

12 feet apart. Find the length of their common chord,

and also that of their common tangent. 9. Two parallel chords of a circle are c and G1 and their

distance apart is d, to find the radius of the

circle. 10. If v is the versed sine of an arc, k the chord of half the

arc, and r the radius, k2=2vr.

177o. Theorem.-If upon each of two intersecting lines a pair of points be taken such that the rectangle on the segments between the points of intersection and the assumed points in one of the lines is equal to the corresponding rect

M

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angle for the other line, the four assumed points are concyclic. B

(Converse of 176o.) L and M intersect in O, and

OA. OB=OC.OD.

Then A, B, C, and D are concyclic. Proof.Since the os are equal, if A and B lie upon the same side of O, C and D must lie upon the same side of 0; and if A and B lie upon opposite sides of O, C and D must lie upon opposite sides of O.

Let a o pass through A, B, C, and let it cut M in a second point E. Then OA. OB=OC.OE.

(176°) But OA. OB=OC.OD.

(hyp.) OD=OE, and as D and E are upon the same side of O they must coincide; .. A, B, C, D are concyclic.

q.e.d.

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178°. Let two circles excluding each other without contact have their centres at A and B, and let C be the point, on their common centre-line, which divides AB so that the difference between the squares on the segments AC and CB is equal to the difference between the squares on the con

terminous radii. Through C
draw the line PCD I to AB,
and from any point P on this
line draw tangents PT and PT'
to the circles.

Join AT and BT'.
Then, by construction,

AC2-BC2=AT2- BT2.

But, since PC is an altitude in the ΔΑΡΒ, , AC2 - BC2=AP2 – BP,

(172°, 1) and

AP2=AT2+ PT?, and

BP2= BT+PT', (169', Cor. 1) whence

PT=PT', and

PT=PT.

Therefore PCD is the locus of a point from which equal tangents are drawn to the two circles.

Def.- This locus is called the radical axis of the circles, and is a line of great importance in studying the relations of two or more circles.

Cor. 1. The radical axis of two circles bisects their common tangents.

Cor. 2. When two circles intersect, their radical axis is their common chord.

Cor. 3. When two circles touch externally, the common tangent at the point of contact bisects the other common tangents.

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179°. The following examples give theorems of some importance.

Ex. I. P is any point without a circle and TT' is the chord of contact (114°, Def.) for the point P. TT' cuts the centre-line PO in Q. Then, PTO being a 7, (110o)

OQ. OP=OT. (169) ::. the radius is a geometric mean between the join of any point with the centre and the perpendicular from the centre upon the chord of contact of the point.

Def.-P and Q are called inverse points with respect to the circle.

Ex. 2. Let PQ be a common direct tangent to the circles having O and Oas centres.

Let OP and O'Q be radii to the points of contact, and let QR be || to 00'. Denote A the radii by r and r'. Then

AC=00'+r-r',

BD=00'-ptri. .:. AC.BD=002- (r-)=QR2-PRP=PQ? (169", Cor. 1)

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Similarly it may be shown that
AD. BC=square on the transverse common tangent.

EXERCISES.

1. The greater of two chords in a circle is nearer the centre

than the other. 2. Of two chords unequally distant from the centre the one

nearer the centre is the greater. 3. AB is the diameter of a circle, and P, Q any two points on

the curve. AP and BQ intersect in C, and AQ and BP in C'. Then

AP.AC+BQ.BC=AC'. AQ+BC'.BP. 4. Two chords of a circle, AB and CD, intersect in O and

are perpendicular to one another. If R denotes the radius of the circle and E its centre,

8R2=AB2+CD2 + 40E?. 5. Circles are described on the four sides of a quadrangle as

diameters. The common chord of any two adjacent circles is parallel to the common chord of the other

two. 6. A circle S and a line L, without one another, are touched

by a variable circle 2. The chord of contact of Z passes through that point of S which is farthest distant

from L. 7. ABC is an equilateral triangle and P is any point on its

circumcircle. Then PA+PB+PC=o, if we consider

the line crossing the triangle as being negative. 8. CD is a chord parallel to the diameter AB, and P is any point in that diameter. Then

PC2+ PDP=PA2+ PB?.

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