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Question Number 21006 by Tinkutara last updated on 10/Sep/17

Let z_1 and z_2 be two distinct complex numbers and let z = (1 − t)z_1 + tz_2 for some real number t with 0 < t < 1. If arg(w) denotes the principal argument of a non-zero complex number w, then (1) ∣z − z_1 ∣ + ∣z − z_2 ∣ = ∣z_1 − z_2 ∣ (2) Arg (z − z_1 ) = Arg (z − z_2 ) (3) determinant (((z − z_1 ),(z^� − z_1 ^� )),((z_2 − z_1 ),(z_2 ^� − z_1 ^� ))) = 0 (4) Arg (z − z_1 ) = Arg (z_2 − z_1 )

Letz1andz2betwodistinctcomplexnumbersandletz=(1−t)z1+tz2forsomerealnumbertwith0<t<1.Ifarg(w)denotestheprincipalargumentofanon−zerocomplexnumberw,then(1)∣z−z1∣+∣z−z2∣=∣z1−z2∣(2)Arg(z−z1)=Arg(z−z2)(3)|z−z1z¯−z¯1z2−z1z¯2−z¯1|=0(4)Arg(z−z1)=Arg(z2−z1)

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