72 The parallel axiom; Euclidean geometry Ch. 5
AsGC‖SDthe triangles[A,D,S]and[A,C,G]are similar, so
|A,C|
|A,D|=
|G,C|
|S,D|
In the similar triangles[B,C,H]and[B,D,S],
|B,C|
|B,D|=
|C,H|
|S,D|
Then
|G,C|
|S,D|
=|C,H|
|S,D|
It follows that|G,C|=|C,H|.
S
A
B
C D
G
H
K
Figure 5.12.Let(A,B,C,D)be a harmonic range, S∈AB and K=S be such that S∈[A,K].
Suppose that CS⊥DS. Then CS and DS are the mid-lines of|ASBand|BSK.
Proof. Let the line through C, parallel toDSmeetSAatGandSBatH.ThenC
is the mid-point ofGandH.AlsoCS⊥SD,SD‖GHsoSC⊥GH. It follows that
the triangles[G,C,S]and[H,C,S]are congruent by the SAS-principle. In particular
|∠GSC|◦=|∠HSC|◦and soSCis the mid-line of|ASB. But also|∠CGS|◦=|∠CHS|◦
and in fact the triangle[S,G,H]is isosceles. Now∠CGSand∠DSKare correspond-
ing angles and∠CHSand∠DSHare alternate angles. It follows that|∠DSK|◦=
|∠DSH|◦and so the mid-line of|BSKisSD.
5.6 Area of triangles, and convex quadrilaterals and polygons
5.6.1 Area of a triangle
Let A,B,C be non-collinear points, and D∈BC,E∈CA,F∈AB points such that
AD⊥BC,BE⊥CA,CF⊥AB. Then
|A,D||B,C|=|B,E||C,A|=|C,F||A,B|.