27930

27930 (Nicolae Mușuroia)

Fie Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_1, z_2} respectiv Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_3} , afixele vârfurilor triunghiului Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1A_2A_3} , înscris în cercul $C(0,1)$. Arătați că triunghiul $A_{1}A_{2}A_{3}$ este echilateral dacă și numai dacă $(z_{1}+z_{2})(z_{2}+z_{3})(z_{3}+z_{1})\not =0$ și ${\frac {1}{z_{1}+z_{2}}}+{\frac {1}{z_{2}+z_{3}}}+{\frac {1}{z_{3}+z_{1}}}=0$.

Soluție. Dacă $A_{1}A_{2}A_{3}$ este triunghi echilateral, afixele punctelor $A_{1},A_{2},A_{3}$ sunt $z_{1},\epsilon z_{1}$ și $\epsilon ^{2}z_{1}$, unde $\epsilon \in \mathbb {C}$ astfel încât $\epsilon ^{2}+\epsilon +1=0$, iar relația din enunț se verifică prin calcul elementar.

Reciproc, condiția din enunț se scrie echivalent

$$(z_{1}+z_{2})(z_{2}+z_{3})+(z_{2}+z_{3})(z_{3}+z_{1})+(z_{3}+z_{1})(z_{1}+z_{2})=0,$$

iar după calcule se rescrie $(z_{1}+z_{2}+z_{3})^{2}=-(z_{1}z_{2}+z_{2}z_{3}+z_{3}z_{1})$.

Întrucât $|z_{1}|=|z_{2}|=|z_{3}|=1$, deducem că

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{z_1+z_2+z_3} = \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} = \frac{z_1z_2+z_2z_3+z_3z_1}{z_1z_2z_3} = -\frac{(z_1+z_2+z_3)^2}{z_1z_2z_3}.}

Trecând la module, rezultă că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z_1+z_2+z_3| = |z_1+z_2+z_3|^2} . Întrucât ortocentrul Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} al triunghiului Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1A_2A_3} are afixul Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h = z_1+z_2+z_3} , obținem Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |h| \in \{0,1\}} .

Dacă Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |h| = 1} , atunci Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H \in C(0,1)} . Rezultă că triunghiul Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1A_2A_3} este dreptunghic și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_1+z_2 = 0} sau Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_2+z_3 = 0} sau Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_3+z_1 = 0} , fals.

Așadar, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |h| = 0} , deci Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = O} , adică triunghiul Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1A_2A_3} este echilateral.