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Unformatted text preview: (i) 0 < r < 1;
(ii) r < 0;
(iii) r > 1.
Case (i). Here r + 1 − r = 1. So
z4 − z 1 z3 − z 2
+ 1−
·
z4 − z 2 z3 − z 1 z4 − z 1 z3 − z 2
·
z4 − z 2 z3 − z 1 = 1. Multiplying both sides by the denominator z4 − z2 z3 − z1  gives after
simpliﬁcation
z4 − z1 z3 − z2  + z2 − z1 z4 − z3  = z4 − z2 z3 − z1 ,
or
(a) AD · BC + AB · CD = BD · AC.
Case (ii). Here 1 + r = 1 − r. This leads to the equation
(b) BD · AC + AD · BC + = AB · CD. Case (iii). Here 1 + 1 − r = r. This leads to the equation
(c) BD · AC + AB · CD = AD · BC. Conversely if (a), (b) or (c) hold, then we can reverse the argument to deduce
that r is a complex number satisfying one of the equations
r + 1 − r = 1, 1 + r = 1 − r, from which we deduce that r is real. 64 1 + 1 − r = r, ...
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This note was uploaded on 12/19/2011 for the course MAS 3105 taught by Professor Dreibelbis during the Fall '10 term at UNF.
 Fall '10
 Dreibelbis

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