Sec. 5.3 Ratio results for triangles 67
Still within the first case, now suppose that|A,P|
|A,B|=x,|A,Q|
|A,C|
=y,wherexis an irrational number with 0<x<1. Ifuis any positive rational number
less thanx,andPuis a point chosen on[A,B]so that
|A,Pu|
|A,B|=u,then the line throughPuwhich is parallel toBCwill meet[A,C]in a pointQusuch
that
|A,Qu|
|A,C|
=u.Similarly ifvis any rational number such thatx<v<1, andPvis a point chosen on
[A,B]so that
|A,Pv|
|A,B|
=v,then the line throughPvwhich is parallel toBCwill meet[A,C]in a pointQvsuch
that
|A,Qv|
|A,C|
=v.As|A,Pu|<|A,P|<|A,Pv|we haveP∈[Pu,Pv]. It follows by 4.3.2 thatQ∈[Qu,Qv]
and sou<y<v. Thus for all rationaluandvsuch thatu<x<vwe haveu<y<v.
It follows thatx=y.
This completes the first case. For the second case note that ifP∈[A,B]we have
B∈[A,P]. Then by the first case
|A,B|
|A,P|=
|A,C|
|A,Q|
,
so the reciprocals of these are equal.
5.3.2 Similar triangles
Let A,B,C and A′,B′,C′be two sets of non-collinear points such that
|∠BAC|◦=|∠B′A′C′|◦,|∠CBA|◦=|∠C′B′A′|◦,|∠ACB|◦=|∠A′C′B′|◦.Then
|B′,C′|
|B,C|
=
|C′,A′|
|C,A|
=
|A′,B′|
|A,B|
.
Thus if the degree-measures of the angles of one triangle are equal, respectively, to
the degree-measures of the angles of a second triangle, then the ratios of the lengths
of corresponding sides of the two triangles are equal.