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If the circles are represented in trilinear coordinates in the usual way, then their radical center is conveniently given as a certain determinant.

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## Some Related Sentences

If and circles

__If__

**the**dominant country's influence

**is**felt

**in**social and cultural

__circles__

**,**such

**as**" foreign " music being popular with young people

**,**it may be described

**as**cultural imperialism

**.**

__If__

**the**parameter " dopefish "

**is**added to

**the**executable

**,**

**a**sample of burping

**is**heard and Scott's Mystical Head

**is**seen spinning

**in**

__circles__on

**the**screen

**.**

__If__you measure

**the**circumferences of

__circles__of steadily larger diameters and divide

**the**former by

**the**latter

**,**all three geometries give

**the**value π for small enough diameters but

**the**ratio departs from π for larger diameters unless Ω = 1:

*

__If__four arbitrary points A**,**B**,**C**,**D**are****given**that do not form an orthocentric system**,****then****the**nine-point__circles__of ABC**,**BCD**,**CDA and DAB concur at**a**point**.**
*

__If__four points A**,**B**,**C**,**D**are****given**that form**a**cyclic quadrilateral**,****then****the**nine-point__circles__of ABC**,**BCD**,**CDA and DAB concur at**the**anticenter of**the**cyclic quadrilateral**.**
(

__If__they**are**not orientable**the**natural fibration by__circles__**is**not necessarily**a**Seifert fibration:**the**problem**is**that some fibers may " reverse orientation ";**in**other words**their**neighborhoods look like fibered solid Klein bottles rather than solid tori .< ref > Ronald Fintushel**,**Local S < sup > 1 </ sup > actions on 3-manifolds**,**Pacific J**.**o**.**M**.**66 No1 ( 1976 ) 111-118**,**http :// projecteuclid**.**org /...) The classification of such ( oriented ) manifolds**is****given****in****the**article on Seifert fiber spaces**.**
*

__If__**a**circle q passes through two distinct points A and A ', inverses with respect to**a**circle k**,****then****the**__circles__k and q**are**orthogonal**.**
*

__If__**the**__circles__k and q**are**orthogonal**,****then****a**straight line passing through**the****center**O of k and intersecting q**,**does so at inverse points with respect to k**.**__If__

**the**

**radical**

**center**lies outside of all three

__circles__

**,**

**then**it

**is**

**the**

**center**of

**the**unique circle (

**the**

**radical**circle ) that intersects

**the**three

**given**

__circles__orthogonally ;

**the**construction of this orthogonal circle corresponds to Monge's problem

**.**

__If__C and C

**are**tangent Ford

__circles__

**,**

**then**

**the**half-circle joining ( p / q

**,**0 ) and ( r / s

**,**0 ) that

**is**perpendicular to

**the**x-axis

**is**

**a**hyperbolic line that also passes through

**the**point where

**the**two

__circles__

**are**tangent to one another

**.**

__If__M consists of

**a**circle

**,**and N of two

__circles__

**,**M and N together make up

**the**boundary of

**a**pair of pants W ( see

**the**figure at right ).

__If__you watch

**the**animation above you will see two

__circles__( one about half

**way**between

**the**edge and

**center**

**,**and

**the**other on

**the**edge itself ) and

**a**straight line bisecting

**the**disk

**,**where

**the**displacement

**is**close to zero

**.**

__If__

**a**straight line

**is**considered

**a**degenerate circle with zero curvature ( and thus infinite radius ), Descartes ' theorem also applies to

**a**line and two

__circles__that

**are**all three mutually tangent

**,**giving

**the**radius of

**a**third circle tangent to

**the**other two

__circles__and

**the**line

**.**

__If__four

__circles__

**are**tangent to each other at six distinct points

**,**and

**the**

__circles__have curvatures k < sub > i </ sub > ( for i = 1

**,**..., 4 ), Descartes ' theorem says:

__If__one of

**the**three

__circles__

**is**replaced by

**a**straight line

**,**

**then**one k < sub > i </ sub >, say k < sub > 3 </ sub >,

**is**zero and drops out of equation ( 1 ).

__If__two

__circles__

**are**replaced by lines

**,**

**the**tangency between

**the**two replaced

__circles__becomes

**a**parallelism between

**their**two replacement lines

**.**

__If__

**the**cloth

**is**first folded

**then**bound

**,**

**the**resulting

__circles__will be

**in**

**a**pattern depending on

**the**fold used

**.**

If and are

__If__it were not for an old professor who made me read

**the**classics I would have been stymied on what to do

**,**and now I understand why they

__are__classics ; ;

__If__I even hint at it do you think it will matter that you

__are__his nephew -- and not even

**a**blood nephew ''??

__If__his dancers

__are__sometimes made to look

**as**if they might be creatures from Mars

**,**this

**is**consistent with his intention of placing them

**in**

**the**orbit of another world

**,**

**a**world

**in**which they

__are__freed of

**their**pedestrian identities

**.**

__If__love reflects

**the**nature of man

**,**

**as**Ortega Y Gasset believes

**,**if

**the**person

**in**love betrays decisively what he

**is**by his behavior

**in**love

**,**

**then**

**the**writers of

**the**beat generation

__are__creating

**a**new literary genre

**.**

Defoe

**then**commented**,**``__If__they Could Draw that young Gentleman into Their Measures They would show themselves quickly**,**for they__are__not asham'd to Say They want only**a**head to Make**a**beginning ''**.**__If__Jews

__are__identified

**as**

**a**religious body

**in**

**a**controversy that comes before

**a**national or international tribunal

**,**it

**is**obviously compatible with

**the**goal of human dignity to protect freedom of worship

**.**

Without preliminaries

**,**Esther asked him**,**``__If__you__are__**a**world citizen**,**will you take Garry Davis' place**in**his tent while he goes to**the**hospital ''??__If__we

__are__to believe

**the**list of titles printed

**in**Malraux's latest book

**,**La Metamorphose Des Dieux

**,**Vol. 1 ( ( 1957 )

**,**he

**is**still engaged

**in**writing

**a**large novel under his original title

**.**

__If__

**the**would-be joiner asks these questions he

**is**not likely to be duped by extremists who

__are__seeking to capitalize on

**the**confusions and

**the**patriotic apprehensions of Americans

**in**

**a**troubled time

**.**

__If__

**the**record buyer's tastes

__are__somewhat eclectic or even

**the**slightest bit esoteric

**,**he will find them satisfied on educational records

**.**

__If__it will simply delay

**the**debates until

**the**qualifications

__are__closed next spring

**,**and

**then**carry all

**the**candidates on

**a**tour of debates

**,**it can provide

**a**service to

**the**state

**.**

__If__they

__are__to be commended for foresight

**in**

**their**planning

**,**what

**then**

**is**

**the**judgment of

**a**town council that compounds this problem during

**the**planning stage??

__If__

**the**Communists

__are__sincere

**in**wanting

**a**united

**,**neutral and disarmed Germany

**,**it might well be advantageous for

**the**German people

**in**this nuclear age

**.**

__If__only this could be done more often -- with such heartening results -- many of

**the**earth's `` big problems '' would shrink to

**the**insignificances they really

__are__

**.**

__If__any

__are__left

**,**presently

**,**we may expect to see signs specifically prohibiting

**the**feeding of them too

**.**

__If__

**the**raw population figures

__are__crucially relevant

**,**

**then**it

**is**idle to think of liberation

**,**

**as**idle

**as**to suppose that Poland might liberate Russia

**.**

__If__it

**is**not enough that all of our internationalist One Worlders

__are__advocating that we join this market

**,**I refer you to an article

**in**

**the**New York Times' magazine section ( Nov. 12

**,**1961 )

**,**by Mr. Eric Johnston

**,**entitled `` We Must Join The Common Market ''

**.**

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