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Numerele complexe distincte \( z_{1} , z_{2} , z_{3} \) sunt afixele varfurilor unui triunghi echilateral daca si numai daca verifica o relatie de forma: \( z_{1} + z_{2}\epsilon +z_{3}\epsilon^2 =0 \), unde \( \epsilon \) e o radacina a unitatii de ordin 3, diferita de 1.

Adriana Nistor wrote: Numerele complexe distincte \( z_{1} \) , \( z_{2} \) , \( z_{3} \) sunt afixele varfurilor unui triunghi echilateral \( A_1A_2A_3 \) daca si numai

daca verifica o relatie de forma \( z_{1} + z_{2}\epsilon +z_{3}\epsilon^2 =0 \), unde \( \epsilon \) e o radacina a unitatii de ordinul trei , diferita de \( 1 \) .

Dem. Notam \( w=\cos\frac {\pi}{3}+i\sin\frac {\pi}{3} \) . Se observa ca \( w^3=-1 \) , \( w^2-w+1=0 \) si \( \overline w=-w^2 \) si radacinile nereale de ordinul \( 3 \) ale lui \( 1 \) sunt

\( \underline{\overline{\left\|\ \epsilon=w^2\ \ \wedge\ \ \epsilon^2=-w\ \right\|}} \) . Asadar \( \triangle A_1A_2A_3 \) este echilateral \( \Longleftrightarrow \) \( z_1-z_2=w(z_3-z_2)\ \ \vee\ \ z_1-z_3=w(z_2-z_3)\ \Longleftrightarrow \)

\( z_1+(w-1)z_2-wz_3=0\ \ \vee\ \ z_1-wz_2+(w-1)z_3=0\ \Longleftrightarrow \) \( z_1+w^2z_2-wz_3=0\ \ \vee\ \ z_1-wz_2+w^2z_3=0\ \Longleftrightarrow \)

\( z_1+\epsilon z_2+\epsilon^2z_3=0\ \ \vee\ \ z_1+\epsilon^2z_2+\epsilon z_3=0 \) . Radacinile de ordinul trei ale unitatii sunt \( 1 \) , \( \epsilon \) , \( \epsilon^2 \) , unde \( \epsilon^3=1 \) si \( \epsilon^2+\epsilon +1=0 \) .

Remarca 1. \( \triangle A_1A_2A_3 \) este echilateral \( \Longleftrightarrow \) \( z_1+w^2z_2-wz_3=0\ \ \vee\ \ z_1-wz_2+w^2z_3=0\ \Longleftrightarrow \)

\( \left(z_1+w^2z_2-wz_3\right)\left(z_1-wz_2+w^2z_3\right)=0\ \Longleftrightarrow\ z_1^2+z_2^2+z_3^2+\left(w^2-w\right)\left(z_1z_2+z_2z_3+z_3z_1\right)=0 \) . Insa

\( w^2-w+1=0 \) . In concluzie, \( \underline{\overline{\left\|\ \triangle\ A_1A_2A_3\ \mathrm{este\ echilateral}\ \Longleftrightarrow\ z_1^2+z_2^2+z_3^2=z_1z_2+z_2z_3+z_3z_1\ \right\|}} \) .

Remarca 2. Consideram unghiul orientat \( \widehat {BAC} \) de la semidreapta \( [AC \) catre semidreapta \( [AB \) in sensul

invers acelor unui ceasornic si care are masura \( \phi\ \in\ \left(0\ ,\ 2\pi\right) \). Daca \( A(a) \) , \( B(b) \) , \( C(c) \) , atunci aceasta

pozitie a unghiului este caracterizata prin relatia \( \underline{\overline{\left\|\ b-a=\rho\cdot\omega (c-a)\ \mathrm{,\ unde}\ \rho =\frac {|b-a|}{|c-a|}\ \mathrm{si}\ \omega =\cos\phi +i\sin\phi\ \right\|}} \) .

Situatii particulare mai des intalnite. \( AB=AC\ \Longrightarrow\ b-a=\omega\cdot (c-a) \) ; \( AB=AC\ \ \wedge\ \ \phi =90^{\circ}\Longrightarrow\ b-a=i\cdot (c-a) \) ;

\( AB=AC\ \ \wedge\ \ \phi =60^{\circ}\Longrightarrow\ b-a=\omega\cdot (c-a) \) , unde \( \omega=\cos\frac {\pi}{3}+i\sin\frac {\pi}{3}\ ,\ w^3=-1\ ,\ w^2-\omega +1=0\ ,\ \overline{\omega}=-\omega^2 \) .