Geometry with Trigonometry
220 Vector and complex-number methods Ch. 11 By (11.6.4) the circumcentreZ′ 16 of the triangle[Z 4 ,Z 5 ,Z 6 ]is given by z′ 16 ...
Sec. 11.6 Mobile coordinates 221 Then p^22 +q^22 = ( 2 p 1 − 1 )^2 (p^21 +q^21 ) (p^21 +q^21 )^2 = ( 2 p 1 − 1 )^2 p^21 +q^21 . ...
222 Vector and complex-number methods Ch. 11 follows. Starting with our usual notationz 1 =z 2 +(p 1 +ıq 1 )(z 3 −z 2 )let us se ...
Sec. 11.6 Mobile coordinates 223 Nowz 4 =z 2 +^12 (z 3 −z 2 ),z 8 =z 2 +p 1 (z 3 −z 2 ), and by the formula in 11.6.5 for the or ...
224 Vector and complex-number methods Ch. 11 Then p^24 +q^24 =p^24 +( (p 1 − 1 ) q 1 )^2 p^24 , p^24 +q^24 q 4 = p^24 q 4 ( 1 +( ...
Sec. 11.6 Mobile coordinates 225 z( 16 iii)=z 2 +p 1 (z 3 −z 2 )+^12 ( 1 − q 1 p 1 ı)[z 2 +^12 p 1 ( 1 − p 1 − 1 q 1 ı)−z 2 −p 1 ...
226 Vector and complex-number methods Ch. 11 11.7 Somewell-knowntheorems...................... NOTE. The advantage of mobile coo ...
Sec. 11.7 Some well-known theorems 227 Recalling from 11.6.5 that the incentreZ 15 has complex coordinate z 15 =z 2 + p 1 + √ p^ ...
228 Vector and complex-number methods Ch. 11 It will also meet the nine-point circle at the point z 22 =z′ 16 − r 1 |z 15 −z′ 16 ...
Sec. 11.7 Some well-known theorems 229 For the nine-point circle we take the pointZ 26 such that z 26 =z′ 16 + r 1 |z 15 −z′ 16 ...
230 Vector and complex-number methods Ch. 11 s(z 1 −z 3 ).Butz 1 −z 3 =(p 1 − 1 +q 1 ı)(z 3 −z 2 ),soz 3 −z 2 =p 1 − 11 +q 1 ı(z ...
Sec. 11.7 Some well-known theorems 231 11.7.3 The incentre on the Euler line of a triangle ........... We suppose that we have t ...
232 Vector and complex-number methods Ch. 11 and this is a non-zero multiple of ∣∣ ∣∣ ∣∣ ∣∣ c^2 +a^2 −b^2 (c^2 +a^2 −b^2 )(a^2 + ...
Sec. 11.7 Some well-known theorems 233 we havew 3 =z 2 +(p+^12 +ıq′)(z 3 −z 2 ), for some real numberq′.ButZ 4 ,W 2 ,W 3 are col ...
234 Vector and complex-number methods Ch. 11 11.8 Isogonal conjugates .......................... 11.8.1 Isogonal conjugates Defi ...
Sec. 11.8 Isogonal conjugates 235 wherev=0asZ 1 ∈Z 2 Z 3. With this notation we have thatu+vı+λ 1 λ (^2) uu 2 −+vıv 2 is real ...
236 Vector and complex-number methods Ch. 11 Exercises 11.1 Show that ifZ 1 =Z 2 , the pointsZon the perpendicular bisector of[ ...
Sec. 11.8 Isogonal conjugates 237 (iii) ( −→ OZ )= −→ OZ, (iv) ( −−→ OZ 1 + −−→ OZ 2 )= −−→ OZ 1 + −−→ OZ 2 , (v) (k −→ ...
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12 Trigonometric functions in calculus COMMENT. In 3.7.1 and 9.3.1 we extended degree-measure to reflex angles, but the only use ...
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