Geometry with Trigonometry
200 Vector and complex-number methods Ch. 11 Then by repeated use of 10.5.3 and 11.4.5 δF(W 1 ,W 2 ,W 3 ) =δF ( v( 1 −w) v−w Z 2 ...
Sec. 11.5 Vector methods in geometry 201 and since the pointsZ 4 ,Z 5 ,Z 6 are not collinear we can equate the coefficients in t ...
202 Vector and complex-number methods Ch. 11 This must be the pointW 3 then. Similarly, on eliminatingZ 0 from the equations for ...
Sec. 11.5 Vector methods in geometry 203 δF(W 1 ,W 2 ,W 3 )is equal to δF ( q(v−u) qv−pu Z 2 + u(q−p) qv−pu Z 6 , q( 1 −v) 1 −qv ...
204 Vector and complex-number methods Ch. 11 Repeating an argument that we used in 7.2.3, suppose now that m and n are any lines ...
Sec. 11.5 Vector methods in geometry 205 for the angles of a triangle. For, using the notation of 10.8.1,|α|◦+|β|◦+|γ|◦=180, and ...
206 Vector and complex-number methods Ch. 11 This circle is called theincirclefor the triangle; its centreZ 15 is called theince ...
Sec. 11.6 Mobile coordinates 207 For anyZ∈Π,Z=O,wedefine −→ OZ ⊥ = −−→ OWwhereZ≡(x,y),W≡(−y,x),and call this theGrassmann suppl ...
208 Vector and complex-number methods Ch. 11 Given non-collinear pointsZ 1 ,Z 2 ,Z 3 , by 11.4.1 we can express −−→ OZ 1 = −−→ O ...
Sec. 11.6 Mobile coordinates 209 Note too that if z−z 2 =(p+qı)(z 3 −z 2 ),z′−z 2 =(p′+q′ı)(z 3 −z 2 ), then |z′−z|=|p′−p+(q′−q) ...
210 Vector and complex-number methods Ch. 11 By Pythagoras’ theorem |z 16 −z 2 |^2 =|z 4 −z 2 |^2 +|z 4 −z 16 |^2 ={^12 |z 3 −z ...
Sec. 11.6 Mobile coordinates 211 By cyclic rotation we can write down the other two coefficients and so have Z 16 = 1 2 cosα sin ...
212 Vector and complex-number methods Ch. 11 Similarly the point with complex coordinatez 2 +|Z 21 ,Z 1 |(z 1 −z 2 ), lies on th ...
Sec. 11.6 Mobile coordinates 213 The mid-point sought is z 3 + 1 2 |Z 2 ,Z 3 | ⎡ ⎣z 2 −z 3 +√^1 −p^1 −ıq^1 ( 1 −p 1 )^2 +q^21 (z ...
214 Vector and complex-number methods Ch. 11 Assumingq 1 =0 we have the pair of equations 1 √ p^21 +q^21 r− 1 √ ( 1 −p 1 )^2 +q ...
Sec. 11.6 Mobile coordinates 215 11.6.6 Euler line of a triangle ....................... With the notationZ 7 ,Z 16 ,Z 11 for th ...
216 Vector and complex-number methods Ch. 11 Next suppose thatz 1 =z 2 +(p 1 +q 1 ı)(z 3 −z 2 ),z′ 1 =z′ 2 +(p 1 −q 1 ı)(z′ 3 −z ...
Sec. 11.6 Mobile coordinates 217 11.6.8 Centroids of similar triangles erected on the sides of a triangle.... Given an arbitrary ...
218 Vector and complex-number methods Ch. 11 Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Figure 11.15. Similar triangles on sides of trian ...
Sec. 11.6 Mobile coordinates 219 Also, by (11.6.12) z( 16 iii)+(p 1 +q 1 ı)(z( 16 ii)−z( 16 iii)) =( 1 −p 1 −q 1 ı)z( 16 iii)+(p ...
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