Geometry with Trigonometry

58 Congruence of triangles; parallel lines Ch. 4

l

πl(P)

P

Figure 4.13. Projection to the linel.

l

sl(P)

P

Axial symmetry in the linel.

Definition. For any linel∈Λwe define a functionsl:Π→Πby specifying that
for allP∈Π,sl(P)is the pointQsuch that

πl(P)=mp(P,Q).

We refer toslasaxial symmetry in the linel.

LetH 1 ,H 2 be closed half-planes with common edge l, let P 1 ∈H 1 \l and P 2 =
sl(P 1 ). Then, for all P∈H 1 ,|P,P 1 |≤|P,P 2 |.
Proof.IfP∈l,then|P,P 1 |=|P,P 2 |, by 4.1.1 whenP∈P 1 P 2 ,andas
P=mp(P 1 ,P 2 )otherwise.
WhenP∈G 1 =H 1 \lwe suppose first thatP∈P 1 P 2 .Then[P,P 2 ]meetslin a
pointQand we have

|P,P 2 |=|P,Q|+|Q,P 2 |=|P,Q|+|Q,P 1 |.

Now we cannot haveQ∈[P,P 1 ]
as[P,P 1 ]⊂G 1 andQ∈l. Thus
either Q∈PP 1 or Q∈PP 1 \
[P,P 1 ].Wethenhave|P,Q|+
|Q,P 1 |>|P,P 1 | by 4.3.1 and
3.1.2. For the case whenP∈
P 1 P 2 , we denote byRthe point
of intersection ofP 1 P 2 andl,so
thatR=mp(P 1 ,P 2 ).

R Q

P

P 1

P 2

l

H 1

H 2

Figure 4.14. Distance and half-planes.

NowP∈[R,P 1 so eitherP∈[R,P 1 ]orP 1 ∈[R,P]. In the first of these cases we
have
|P 1 ,P|<|P 1 ,R|=|R,P 2 |<|P,P 2 |,

asR∈[P,P 2 ]. In the second case we have|P,P 1 |<|P,R|<|P,P 2 |asR∈[P,P 2 ].

Exercises

4.1 IfD=Ais in[A,B,C]but not in[B,C],then|B,D|+|D,C|<|B,A|+|A,C|

and|∠BDC|◦>|∠BAC|◦.