Geometry with Trigonometry

114 Circles; their basic properties Ch. 7

of the reflex- angle with support|∠POQ. By addition we then have that the degree-
measure of this reflex-angle is equal to 2|∠PRQ|◦.

Definition. If the vertices of a convex quadrilateral all lie on some circle, then the
quadrilateral is said to becyclic.

COROLLARY.Let[P,Q,R,S]be a convex cyclic quadrilateral. Then the sum of
the degree-measures of a pair of opposite angles is 180.
Proof. Using the fourth diagram in Fig. 7.7, we first we note that

|∠RPQ|◦+|∠PQR|◦+|∠QRP|◦= 180.

Next as[S,Q⊂IR(|PSR),wehave|∠PSR|◦=|∠PSQ|◦+|∠QSR|◦.But
|∠PSQ|◦=|∠QRP|◦,|∠QSR|◦=|∠RPQ|◦.Hence

|∠PSR|◦+|∠PQR|◦=|∠RPQ|◦+|∠PQR|◦+|∠QRP|◦= 180.

7.5.2 Minorandmajorarcsofacircle....................

Definition.LetP 1 ,P 2 ∈C(O;k)
be distinct points such that
O∈P 1 P 2 .LetH 5 ,H 6 be the
closed half-planes with edge
P 1 P 2 , with O ∈ H 5 .Then
C(O;k)∩H 5 , C(O;k)∩H 6 ,
are called, respectively, thema-
jorandminor arcsofC(O;k)
withend-pointsP 1 andP 2.

O

P P^1

2

W

P

H 5

H 6

Figure 7.8.

The point P∈C(O;k)is in the minor arc with end-points P 1 ,P 2 if and only if
[O,P]∩[P 1 ,P 2 ]=0./
Proof.LetPbe in the minor arc. ThenO∈H 5 ,P∈H 6 so[O,P]meetsP 1 P 2 in
some pointW.AsW∈[O,P]we have|O,W|≤kso by 7.1.1W∈[P 1 ,P 2 ].
Conversely suppose thatW∈[P 1 ,P 2 ]so that|O,W|≤k. ChooseP∈[O,Wso that
|O,P|=k.Thenas|O,W|≤|O,P|we haveW∈[O,P]so that asO∈H 5 we have
P∈H 6.

7.6 Senseddistances............................

7.6.1 Senseddistance ............................

Definition.Iflis a line,≤lis a natural order onlandZ 1 ,Z 2 ∈l, then we define
Z 1 Z 2 ≤lby

Z 1 Z (^2) ≤l=

{

|Z 1 ,Z 2 |, ifZ 1 ≤lZ 2 ,

−|Z 1 ,Z 2 |, ifZ 2 ≤lZ 1 ,

and call this thesensed distancefromZ 1 toZ 2. In knowing this rather than just
the distance fromZ 1 toZ 2 we have extra information which can be turned to good