Geometry with Trigonometry
20 Preliminaries Ch. 1 1.6.2 Classical algebra We need a knowledge of the real number system and the complex number system, and ...
2 Basic shapes of geometry COMMENT. Geometry deals with our intuitions as to the physical space in which we exist, with the prop ...
22 Basic shapes of geometry Ch. 2 or l∩m is a singleton, or l=m and in this last case l∩m=l=m. Moreover the planeΠis not a line, ...
Sec. 2.1 Lines, segments and half-lines 23 2.1.3 Reciprocalorders IfA≤lBwe also writeB≥lAand read this as ‘Bsucceeds or coincide ...
24 Basic shapes of geometry Ch. 2 while if B≤lA so that A≥lB, (2.1.2) we define [B,A]={P∈l:B≤lP≤lA}, [A,B]={P∈l:A≥lP≥lB}. We sho ...
Sec. 2.1 Lines, segments and half-lines 25 (iv) LetC,D∈[A,B]so thatA≤lC≤lBandA≤lD≤lB.ByA 2 (iv) eitherC≤lD orD≤lC. IfC≤lDandP∈[C ...
26 Basic shapes of geometry Ch. 2 2.2 Open and closed half-planes 2.2.1 Convexsets Definition. A setEis said to beconvexif for e ...
Sec. 2.2 Open and closed half-planes 27 (a)G 1 ⊂G 1 ′ or (b)G 1 ⊂G 2 ′ or (c)G 1 ∩G 1 ′= 0 /,G 1 ∩G 2 ′=0./ In (c) we haveP∈G ...
28 Basic shapes of geometry Ch. 2 CASE (b). LetA,B∈l.Then[A,B]⊂l⊂H 1 , so the result follows. CASE (c). Let one ofA,Bbe onland t ...
Sec. 2.3 Angle-supports, interior and exterior regions, angles 29 Definition. We call a pair{[A,B,[A,C}of co-initial half-lines ...
30 Basic shapes of geometry Ch. 2 2.3.2 Exteriorregions Definition.If|BACis an angle-support which is not straight andIR(|BAC)is ...
Sec. 2.4 Triangles and convex quadrilaterals 31 with edgeCAin whichBlies, andH 5 is the closed half-plane with edgeABin whichCli ...
32 Basic shapes of geometry Ch. 2 (iv) Each of the sides[B,C],[C,A],[A,B]is a subset of[A,B,C]. Proof. (i) As∩is commutative,H 1 ...
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 ...
34 Basic shapes of geometry Ch. 2 2.4 If[A,B],[C,D]are both segments of a linelsuch that[A,B]∩[C,D]=0, show/ that[A,B]∩[C,D]and ...
3 Distance; degree-measure of an angle COMMENT. In this chapter we introduce distance as a primitive concept, relate it to the p ...
36 Distance; degree-measure of an angle Ch. 3 contains infinitely many points and this supersedes the specification in A 1 thatl ...
Sec. 3.1 Distance 37 (iv) We haveA=Bby definition of[A,B,andA=Cas 0<|A,B|<|A,C|so that 0<|A,C|.WealsohaveB=Cas|A,B|& ...
38 Distance; degree-measure of an angle Ch. 3 A=C D B Figure 3.3. A C B DX 3.2 Mid-points............................... ...
Sec. 3.3 A ratio result 39 (ii)For all A,B∈Π,mp(A,B)∈[A,B]. (iii)In all cases mp(A,A)=A, and mp(A,B)=A, mp(A,B)=BwhenA=B. (iv ...
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