Sec. 7.5 Angles standing on arcs of circles 113
and of course (c) is ruled out by assumption.
S
R
P
O
Q
U
P QR
O
V
P Q
R
O
W
V
Figure 7.7.Q
PR
S
We suppose that (a) holds as in the first figure; the case of (b) is treated similarly.
Then there is a pointU∈[P,S]∩QR.AsU∈[P,S],Umust be an interior point for
the circle and hence we must haveU∈[Q,R]. By 7.4.2
|U,P|
|U,Q|=
|U,R|
|U,S|
,
andwealsohave|∠PUR|◦=|∠QUS|◦as these are opposite angles. By 5.3.2, the
triangles[U,P,R],[U,Q,S]are similar. In particular|∠PRU|◦=|∠QSU|◦.Thefirst
diagram in Fig. 7.7 deals with this general case.
In (i) whenO∈PQ, that we have a right-angle comes from 7.2.1 and there is a
diagram for this in Fig. 7.2.
WhenO∈PQwe letVbe the mid-point of{P,Q}and for the second case (ii), as in
the second diagram in Fig. 7.7, we takeRto be point in which[V,Omeets the circle.
Then by 5.2.2, Corollary, and 4.1.1(i)|∠VOP|◦= 2 |∠VRP|◦,|∠VOQ|◦= 2 |∠VRQ|◦.
But[R,V ⊂IR(|PRQ)so that|∠VRP|◦
+|∠VRQ|◦=|∠PRQ|◦. Moreover[O,V ⊂IR(|POQ)so that|∠VOP|◦+|∠VOQ|◦
=|∠POQ|◦. By addition we then have that|∠POQ|◦= 2 |∠PRQ|◦.
For the third case (iii), as in the third diagram in Fig. 7.7, we takeRto be point
in which[O,V meets the circle andW=Oa point such thatO∈[V,W].Thenby
5.2.2, Corollary, and 4.1.1(i)|∠WOP|◦= 2 |∠WRP|◦,|∠WOQ|◦= 2 |∠WRQ|◦.But
[R,W ⊂IR(|PRQ)so that|∠WRP|◦+|∠WRQ|◦=|∠PRQ|◦. Moreover[O,V ⊂
IR(|POQ)so that by 3.7.1|∠WOP|◦+|∠WOQ|◦is equal to the degree-measure