Geometry with Trigonometry
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5 The parallel axiom; Euclidean geometry COMMENT. The effect of introducing any axiom is to narrow things down, and de- pending ...
62 The parallel axiom; Euclidean geometry Ch. 5 Proof. ChooseR=Pso thatP∈[T,R].ThenR∈nandBandRare on opposite sides ofAP,sothat ...
Sec. 5.2 Parallelograms 63 asAC=BDwould implyB∈AC. SimilarlyC∈IR(|BAD)soTis on[B,D]. Thus [A,C]∩[B,D]=0 so as in 2.4.4 a convex ...
64 The parallel axiom; Euclidean geometry Ch. 5 Thus the sum of the degree-measures of the wedge-angles of a triangle is equal t ...
Sec. 5.3 Ratio results for triangles 65 Proof. By Pasch’s property in 2.4.3 applied to the triangle [A,B,C], the lines through [ ...
66 The parallel axiom; Euclidean geometry Ch. 5 Let A,B,C be non-collinear points and let P∈[A,B and Q∈[A,C be such that PQ‖BC. ...
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 a ...
68 The parallel axiom; Euclidean geometry Ch. 5 A′ B′ C′ A B C B′′ C′′ Figure 5.7. Similar triangles. textProof. ChooseB′ ...
Sec. 5.4 Pythagoras’ theorem, c. 550B.C. 69 Now the line throughB′′which is parallel toBCwill meet[A,C in a pointDsuch that |A,B ...
70 The parallel axiom; Euclidean geometry Ch. 5 CONVERSEofPythagoras’ theorem.Let A,B,C be non-collinear points such that |B,C|^ ...
Sec. 5.5 Mid-lines and triangles 71 It follows that|∠EAC|◦=|∠ACH|◦,andasE,Hare on opposite sides ofAC this implies thatAE‖HC. It ...
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| I ...
Sec. 5.6 Area of triangles, and convex quadrilaterals and polygons 73 A B C E F A B C E F D Figure 5.13. Proof. The ...
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 perpe ...
Sec. 5.6 Area of triangles, and convex quadrilaterals and polygons 75 First suppose thatD=B.Then|∠ABC|◦= 90 =|∠A′B′C′|◦and soD′= ...
76 The parallel axiom; Euclidean geometry Ch. 5 We cannot haveA=Tas then we would haveA∈BD; nor canTbe any other point of the li ...
Sec. 5.6 Area of triangles, and convex quadrilaterals and polygons 77 P 1 P 2 V P 3 U P 1 P 2 P 3 U P 4 ...
78 The parallel axiom; Euclidean geometry Ch. 5 [P 1 ,U meets[P 2 ,P 3 ]at a pointV=P 2 so thatU∈[P 1 ,V].Also[P 1 ,U meets[Pn− ...
Sec. 5.6 Area of triangles, and convex quadrilaterals and polygons 79 ThenPQ‖BC. 5.6 LetAB⊥ACand letD=mp(B,C). Prove that|D,A|=| ...
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