Sec. 7.2 Properties of circles 105
Similar results hold whenX∈[M,Q, that is the points of[M,Q]{Q}are interior
to the circle while the points of([M,Q)[M,Q]are exterior to the circle. ButM∈
[P,Q]so that[P,M]∪[M,Q]=[P,Q],[M,P∪[M,Q =PQand so we can take these
results together. Thus the points of[P,Q], other thanPandQ, are interior to the circle,
and the points ofPQ[P,Q]are exterior to the circle, leaving just the pointsPandQ
of the linel=PQin the circle.
(iii) Suppose that|O,M|>k,sothatMis exterior to the circle. ThenM∈land
OM⊥l.IfX∈l,X=M, then by 4.3.1,|O,X|>|O,M|>k,sothatXis exterior to
the circle.
Definition.Iflis a line such thatl∩C(O;k)={P}for a pointP,thenlis called a
tangenttoC(O;k)atP,andPis called the point of contact.Ifl∩C(O;k)={P,Q}
for distinct pointsPandQ,thenlis called asecantforC(O;k)and the segment
[P,Q]is called achordof the circle; whenO∈l=PQ, the chord[P,Q]is called a
diameterof the circle; in that caseO=mp(P,Q).Ifl∩C(O;k)=0, then/ lis called
anon-secant linefor the circle.
NOTE. By the above every point of a tangent to a circle, other than the point of
contact, is an exterior point. If[P,Q]is a chord, every point of the chord other than
its end-pointsPandQis an interior point, while every point ofPQ[P,Q]is exterior.
Every point of a non-secant line is an exterior point.
7.2 Properties of circles ..........................
7.2.1
Circles have the following properties:-
(i)If[Q,S]is a diameter of the circleC(O;k)and P any point of the circle other
than Q and S, then PQ⊥PS.(ii) If points P,Q,S are such that PQ⊥PS, then P is on a circle with diameter
[Q,S].(iii) If P is any point of the circleC(O;k),[Q,S]is any diameter and U=πQS(P),
then U∈[Q,S]and|Q,U|≤ 2 k.(iv) If Q is a point of a circle with centre O and l is the tangent to the circle at Q,
then every point of the circle lies in the closed half-plane with edge l in which
O lies.