Geometry with Trigonometry

Sec. 2.4 Triangles and convex quadrilaterals 33

If we write

D

/ \

AC

\ /

B

then two vertices in[A,B,C,D]will be adjacent if the letters for them in this diagram
are linked.

Exercises

2.1 LetPbe a fixed point inΠ. Identify the union of all linesl∈Λsuch thatP∈l.

2.2 Prove that segments have the following properties:-

(i) IfC∈[A,B],then[A,C]∪[C,B]=[A,B]and[A,C]∩[C,B]={C}.

(ii) IfC∈[A,B]andB∈[A,C]thenB=C.

(iii) IfC∈[A,B]andD∈[A,C],thenC∈[D,B].

(iv) IfB=A,B∈[A,C]andB∈[A,D], then eitherC∈[B,D]orD∈[B,C].

2.3 Prove that half-lines have the following properties:-

(i) IfB∈ρ(l,A,≤l),thenρ(l,B,≤l)⊂ρ(l,A,≤l).

(ii) IfB∈ρ(l,A,≤l),thenρ(l,A,≤l)=[A,B]∪ρ(l,B,≤l)and

[A,B]∩ρ(l,B,≤l)={B}.

(iii) LetB∈ρ(l,A,≤l),A=BandA∈[B,C].ThenC∈ρ(l,A,≤l)only if

C=A.

(iv) LetB∈ρ(l,A,≤l)andA=B.ThenC∈ρ(l,A,≤l)if and only if either

B∈[A,C]orC∈[A,B].

(v) In all cases

ρ(l,A,≤l)∪ρ(l,A,≥l)=landρ(l,A,≤l)∩ρ(l,A,≥l)={A}.

(vi) LetB∈ρ(l,A,≤l)andA=B.ThenC∈ρ(l,A,≥l)if and only ifA∈

[B,C].

(vii) LetB∈ρ(l,A,≤l)andA=B.Then

ρ(l,A,≤l)∪ρ(l,B,≥l)=l,ρ(l,A,≤l)∩ρ(l,B,≥l)=[A,B],

ρ(l,A,≥l)∩ρ(l,B,≤l)= 0 /,ρ(l,A,≥l)∪ρ(l,B,≤l)∪[A,B]=l.

(viii) IfA=B,A=CandC∈[A,B,then[A,B=[A,C.