7 Circles; their basic properties
Hitherto our sets have involved lines and half-planes, and specific subsets of these.
Now we introduce circles and study their relationships to lines. We do not do this just
to admire the circles, and to behold their striking properties of symmetry. They are
the means by which we control angles, and simplify our work on them.
7.1 Intersectionofalineandacircle ...................
7.1.1 .....................................
Definition.IfOis any point of the planeΠandkis any positive real number, we call
the setC(O;k)of all pointsXinΠwhich are at a distancekfromO,i.e.C(O;k)=
{X∈Π:|O,X|=k},thecirclewithcentreOandlength of radiusk.IfX∈C(O;k)
the segment[O,X]is called aradiusof the circle. Any pointUsuch that|O,U|<k
is said to be aninterior pointfor this circle. Any pointVsuch that|O,V|>kis said
to be anexterior pointfor this circle.
For every circleC(O;k)and line l, one of the following holds:-(i)l∩C(O;k)={P}for some point P, in which case every point of l\{P}is
exterior to the circle.(ii) l∩C(O;k)={P,Q}for some points P and Q, with P=Q, in which case every
point of[P,Q]\{P,Q}is interior to the circle, and every point of PQ\[P,Q]is
exterior to the circle.(iii) l∩C(O;k)=0,/ in which case every point of l is exterior to the circle.Proof.LetM=πl(O),andletmbe the line which containsMand is perpendicular
tol,sothatO∈m.
Geometry with Trigonometry
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