28315

28315 (Vasile Pop și 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 P_1P_2\ldots P_n} 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 (n \geq 3)} un poligon regulat 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 M} un punct în interiorul poligonului. Notăm cu 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 M_1} , $M_{2},\ldots ,M_{n}$ simetricele punctului $M$ față de laturile poligonului. Arătați că, pentru orice alegere a punctului $M$, poligoanele $M_{1}$$M_{2}\ldots M_{n}$ au același centru de greutate.

Soluție:

Vom demonstra următoarea lemă: În planul complex, simetricul punctului $M(m)$ față de dreapta determinată de punctele $A(a)$ și $B(b)$, unde $|a|=|b|=1$, este punctul $M^{\prime }$ de afix $m^{\prime }=a+b-ab{\overline {m}}.$
Într-adevăr, din faptul că mijlocul $N(n)$ al segmentului $[MM^{\prime }]$ aparține dreptei $AB$, rezultă că ${\frac {n-a}{b-a}}\in \mathbb {R}$, adică ${\frac {n-a}{b-a}}$ = ${\frac {{\overline {n}}-{\overline {a}}}{{\overline {b}}-{\overline {a}}}},(1),$ iar din $MM^{\prime }\perp AB$, deducem că ${\frac {m^{\prime }-m}{b-a}}\in i\mathbb {R^{*}}$, adică ${\frac {m^{\prime }-m}{b-a}}=-{\frac {{\overline {m^{\prime }}}-{\overline {m}}}{{\overline {b}}-{\overline {a}}}},(2)$. Având în vedere că ${\overline {a}}={\frac {1}{a}},{\overline {b}}={\frac {1}{b}}$ și $n={\frac {m+m^{\prime }}{2}}$, din relația $(1)$ rezultă că

$$m^{\prime }+m=2(a+b)-ab({\overline {m^{\prime }}}+{\overline {m}}),(3)$$

iar din relația $(2)$ că $m^{\prime }-m=ab({\overline {m^{\prime }}}-{\overline {m}}),(4).$ Adunând egalitățile $(3)$ 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 (4)} 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 m^{\prime}=a+b-ab\overline{m}} .

Revenind la problemă, considerăm un reper cartezian cu originea în centrul poligonului, astfel încât afixele punctelor 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 P_n} 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 P_1} să 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 1} , 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 \epsilon = \cos\frac{2\pi}{n}+i\sin\frac{2\pi}{n}} . Ca urmare, afixul punctului 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 P_k} este 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 \epsilon^k} , pentru orice 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 k \in \{1, 2, \ldots, n\} } .

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 m} afixul punctului 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 M} 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 m_k} afixul punctului 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 M_k, 1 \leq k \leq n.} Folosind lema, 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 m_k=\epsilon^k+\epsilon^{k+1}-\epsilon^{2k+1} \overline{m}} , pentru orice 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 k} . În consecință,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 \sum_{k=1}^{n}m_k=(1+\epsilon) \sum_{k=1}^{n}\epsilon^k+\overline{m} \cdot \sum_{k=1}^{n}\epsilon^{2k+1}=(1+\epsilon)\cdot \epsilon \cdot \frac{\epsilon^n-1}{\epsilon-1}+\overline{m}\cdot\epsilon^3\cdot\frac{\epsilon^{2n}-1}{\epsilon^2-1}=0 } deci centrul de greutate al poligonului 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 M_1M_2 \ldots M_n} este originea, indiferent de alegerea punctului 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 M} .