Sec. 4.3 Properties of triangles and half-planes 57
Proof.
Existence.LetA,Bbe distinct points ofl. Take a pointQon the opposite side ofl
fromPand such that|∠BAQ|◦=|∠BAP|◦.AlsotakeR∈[A,Qso that|A,R|=|A,P|.
AsPandRare on opposite sides ofl,[P,R]meetslin a pointS.
We first suppose thatA∈PRso thatA=S.Then[A,P,S]and[A,R,S]are congruent
by the SAS-principle, so in particular|∠ASP|◦=|∠ASR|◦.AsS∈[P,R]it follows that
these are right-angles and soPR⊥l.
In the second case suppose thatA∈PRso thatA=S.ThenS∈[P,R]and by
construction|∠BSR|◦=|∠BSP|◦. Again these are right-angles soPR⊥l.
Uniqueness. Suppose that there are distinct pointsS,T∈lsuch thatPS⊥l,PT⊥
l. ChooseU=Tso thatT∈[S,U].Then|∠UTP|◦=|∠USP|◦=90 and this contra-
dicts 4.2.1.
COMMENT. We refer to this last asdropping a perpendicularfrom the pointP
to the linel.
Let A,B,C be non-collinear points such that AB⊥AC and let D be the foot of the
perpendicular from A to BC. Then D∈[B,C],D=B,D=C.
Proof. By 4.2.1, in a
right-angled triangle each
of the other two angles
have degree-measure less
than 90. By 4.3.1 it then
follows that the side op-
posite the right-angle is
longer than each of the
other sides. It follows that
|B,D|<|A,B|<|B,C|.
By a similar argument
|C,D|<|B,C|.
A
BD C
Figure 4.12.We cannot then haveB∈[C,D]as that would imply|C,B|≤|C,D|,andsim-
ilarly we cannot haveC∈[B,D]with as that would imply|B,C|≤|B,D|. Hence
D∈[B,C],D=B,D=C.
4.3.4 Projectionandaxialsymmetry
Definition. For any linel∈Λwe define a functionπl:Π→lby specifying that
for allP∈Π, πl(P)is the foot of the perpendicular fromPtol. We refer toπlas
projection to the linel.