Geometry with Trigonometry
160 Complex coordinates; sensed angles; angles between lines Ch. 10 10.7 Orientation of a triple of non-collinear points ..... 1 ...
Sec. 10.7 Orientation of a triple of non-collinear points 161 and so z′ 3 −z′ 2 =(z ̄ 3 −z ̄ 2 )cis 2α,z′ 4 −z′ 2 =(z ̄ 4 −z ̄ 2 ...
162 Complex coordinates; sensed angles; angles between lines Ch. 10 ProofWe use the notation of 10.6.1. In the case (10.6.1)z′=z ...
Sec. 10.8 Sensed angles of triangles, the sine rule 163 For non-collinear points Z 1 ,Z 2 ,Z 3 if α=FZ 2 Z 1 Z 3 ,β=FZ 3 Z 2 Z ...
164 Complex coordinates; sensed angles; angles between lines Ch. 10 10.9 Someresultsoncircles ........................ 10.9.1 A ...
Sec. 10.9 Some results on circles 165 10.9.2 A sufficient condition to lie on a circle ............... Let Z 1 ,Z 2 be fixed dis ...
166 Complex coordinates; sensed angles; angles between lines Ch. 10 When this holds and Z and Z 2 are on the same side of Z 3 Z ...
Sec. 10.9 Some results on circles 167 This is an identity due to Euler and from it (z−z 4 )(z 2 −z 3 ) (z−z 3 )(z 2 −z 4 ) + (z− ...
168 Complex coordinates; sensed angles; angles between lines Ch. 10 10.10 Anglesbetweenlines ......................... 10.10.1Mo ...
Sec. 10.10 Angles between lines 169 10.10.3Duo-angles .............................. Whenl 1 ,l 2 are distinct lines, intersecti ...
170 Complex coordinates; sensed angles; angles between lines Ch. 10 We extend our frame of referenceFby taking in connection wit ...
Sec. 10.10 Angles between lines 171 10.10.5 Addition of duo-angles in standard position ..... To deal with addition of duo- angl ...
172 Complex coordinates; sensed angles; angles between lines Ch. 10 is equal to k^2 [(x 5 −x 4 )(y 5 −y 4 )+x 5 y 4 +x 4 y 5 ]−( ...
Sec. 10.10 Angles between lines 173 10.10.6 Addition formulae for tangents of duo-angles ..... (i) We first note that ifαd,βd∈DA ...
174 Complex coordinates; sensed angles; angles between lines Ch. 10 so that y 6 x 6 = 2 x 4 (k−y 4 ) (k−y 4 )^2 −x^24 Then y 6 x ...
Sec. 10.10 Angles between lines 175 10.10.8 Group properties of duo-angles; sensed duo-angles ......... We note the following pr ...
176 Complex coordinates; sensed angles; angles between lines Ch. 10 the terminology used being ‘cross’. Sensed duo-angles were a ...
Sec. 10.11 A case of Pascal’s theorem, 1640 177 for somet=0inR. By 9.4.1 the left-hand side is equal to cosφ 2 −cosθ 1 +ı(sinφ ...
178 Complex coordinates; sensed angles; angles between lines Ch. 10 Figure 10.16. Very symmetrical cases. Proof. For from the fi ...
Sec. 10.11 A case of Pascal’s theorem, 1640 179 and so our condition for transitivity is got by equating the two right- hand sid ...
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