74 The parallel axiom; Euclidean geometry Ch. 5
A
B PDC
A
B C
T
D
Figure 5.14.
Proof.
(i) ForDis the foot of the perpendicular from the vertexAto the opposite side-line
in each of the triangles[A,B,P]and[A,P,C], so withp 1 =|A,D|we have
Δ[A,B,P]=^12 p 1 |B,P|,Δ[A,P,C]=^12 p 1 |P,C|,and the sum of these is
1
2 p^1 (|B,P|+|P,C|)=1
2 p^1 |B,C|,asP∈[B,C].
(ii) As in 5.2.1 denote byTthe point which[A,C]and[B,D]have in common.
Then by (i) above,
Δ[A,B,D]+Δ[C,B,D]
=(Δ[A,B,T]+Δ[A,D,T])+(Δ[C,B,T]+Δ[C,D,T]),
Δ[A,B,C]+Δ[A,D,C]
=(Δ[A,B,T]+Δ[C,B,T])+(Δ[A,D,T]+Δ[C,D,T]),and these are clearly equal.
(iii)
LetTbe a triangle with vertices{A,B,C}andT′a triangle with vertices
{A′,B′,C′}such thatT(A,B,C)→≡(A′,B′,C′)T′. Suppose thatDandD′are the feet of the
perpendiculars fromAandA′ontoBCandB′C′, respectively.
There are three cases to be considered.
A
B=D C
A
B D C
Figure 5.15.