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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 )

$$\mathrm{Let}\:{z}_{\mathrm{1}} \:\mathrm{and}\:{z}_{\mathrm{2}} \:\mathrm{be}\:\mathrm{two}\:\mathrm{distinct}\:\mathrm{complex} \\ $$ $$\mathrm{numbers}\:\mathrm{and}\:\mathrm{let}\:{z}\:=\:\left(\mathrm{1}\:−\:{t}\right){z}_{\mathrm{1}} \:+\:{tz}_{\mathrm{2}} \:\mathrm{for} \\ $$ $$\mathrm{some}\:\mathrm{real}\:\mathrm{number}\:{t}\:\mathrm{with}\:\mathrm{0}\:<\:{t}\:<\:\mathrm{1}.\:\mathrm{If} \\ $$ $$\mathrm{arg}\left({w}\right)\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{principal}\:\mathrm{argument} \\ $$ $$\mathrm{of}\:\mathrm{a}\:\mathrm{non}-\mathrm{zero}\:\mathrm{complex}\:\mathrm{number}\:{w},\:\mathrm{then} \\ $$ $$\left(\mathrm{1}\right)\:\mid{z}\:−\:{z}_{\mathrm{1}} \mid\:+\:\mid{z}\:−\:{z}_{\mathrm{2}} \mid\:=\:\mid{z}_{\mathrm{1}} \:−\:{z}_{\mathrm{2}} \mid \\ $$ $$\left(\mathrm{2}\right)\:\mathrm{Arg}\:\left({z}\:−\:{z}_{\mathrm{1}} \right)\:=\:\mathrm{Arg}\:\left({z}\:−\:{z}_{\mathrm{2}} \right) \\ $$ $$\left(\mathrm{3}\right)\:\begin{vmatrix}{{z}\:−\:{z}_{\mathrm{1}} }&{\bar {{z}}\:−\:\bar {{z}}_{\mathrm{1}} }\\{{z}_{\mathrm{2}} \:−\:{z}_{\mathrm{1}} }&{\bar {{z}}_{\mathrm{2}} \:−\:\bar {{z}}_{\mathrm{1}} }\end{vmatrix}\:=\:\mathrm{0} \\ $$ $$\left(\mathrm{4}\right)\:\mathrm{Arg}\:\left({z}\:−\:{z}_{\mathrm{1}} \right)\:=\:\mathrm{Arg}\:\left({z}_{\mathrm{2}} \:−\:{z}_{\mathrm{1}} \right) \\ $$

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