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'''Enunț''' | '''Enunț''' | ||
''Fie triunghiul echilateral <math>ABC</math> înscris în cercul de centru <math>O</math> și rază <math>1</math>. Considerăm mulțimea <math>\mathcal{M}</math> a punctelor <math>X</math> din plan cu proprietatea că | ''Fie triunghiul echilateral <math>ABC</math> înscris în cercul de centru <math>O</math> și rază <math>1</math>. Considerăm mulțimea <math>\mathcal{M}</math> a punctelor <math>X</math> din plan cu proprietatea că <math>\overrightarrow{OX} = k \cdot \overrightarrow{OA} + m \cdot \overrightarrow{OB} + n \cdot \overrightarrow{OC}</math>, unde <math>k, m, n \in N^*</math>. Arătați că oricare ar fi punctele distincte <math>M, N, P \in \mathcal{M} </math> există <math>Q\in\mathcal{M}</math> astfel încât vectorii <math>\overrightarrow{MN}</math>, <math>\overrightarrow{PQ} </math> și <math>\overrightarrow{NM}+</math> <math>\overrightarrow{QP}</math> să formeze un triunghi echilateral.'' | ||
'''Soluție''' | '''Soluție''' | ||
Formăm în plan o rețea de triunghiuri echilaterale ale căror vârfuri se află pe drepte paralele echidistante, având direcțiile dreptelor OA,OB și OC, | Formăm în plan o rețea de triunghiuri echilaterale ale căror vârfuri se află pe drepte paralele echidistante, având direcțiile dreptelor <math>OA</math>, <math>OB</math> și <math>OC</math>, distanța ditre două drepte consecutive fiind de <math>\sqrt{3}</math>. Cum <math>\overrightarrow{OC}</math> = <math>-\overrightarrow{OA} - \overrightarrow{OB}</math> obținem că <math>X\in \mathcal{M}</math> dacă și numai dacă există <math>p, q\in \mathbb{Z}</math>, astfel încât <math>\overrightarrow{OX} = p\cdot\overrightarrow{OA} + q\cdot\overrightarrow{OB} </math>. | ||
distanța ditre două drepte consecutive fiind de <math>\sqrt{3}</math> | |||
Analog,coordonatele lui <math>\overrightarrow{OX}</math> în baza <math>(\overrightarrow{OB},\,\overrightarrow{OC})</math>, precum și cele din baza <math>(\overrightarrow{OA},\,\overrightarrow{OC})</math> sunt întregi. De aici rezultă ușor că <math>M</math> este mulțimea tuturor vârfurilor rețelei. | Analog, coordonatele lui <math>\overrightarrow{OX}</math> în baza <math>(\overrightarrow{OB},\,\overrightarrow{OC})</math>, precum și cele din baza <math>(\overrightarrow{OA},\,\overrightarrow{OC})</math> sunt întregi. De aici rezultă ușor că <math>\mathcal{M}</math> este mulțimea tuturor vârfurilor rețelei. | ||
Alegem punctele <math>M,N,P \in M </math>. Dacă vectorul <math>\overrightarrow{MN}</math> este paralel cu unul dintre vectorii <math>\overrightarrow{OA}</math>,<math>\overrightarrow{OB}</math>,<math>\overrightarrow{ | Alegem punctele <math>M, N, P \in \mathcal{M} </math>. Dacă vectorul <math>\overrightarrow{MN}</math> este paralel cu unul dintre vectorii <math>\overrightarrow{OA}</math>, <math>\overrightarrow{OB}</math>, <math>\overrightarrow{OC}</math>, atunci problema este evidentă. Dacă <math>\overrightarrow{MN}</math> nu este paralel cu niciunul dintre vectorii <math>\overrightarrow{OA}</math>, <math>\overrightarrow{OB}</math>, <math>\overrightarrow{OC}</math>, atunci fie <math>a, b\in \mathbb{Z}^*</math> coordonatele vectorului <math>\overrightarrow{MN}</math> în baza <math>(\overrightarrow{OA}, \overrightarrow{OB})</math> și punctele <math>S, T, R</math> astfel încât <math>\overrightarrow{SN} = a \cdot \overrightarrow{OA}</math>, <math>\overrightarrow{MS} = b \cdot \overrightarrow{OB}</math>, <math>\overrightarrow{NR} = a \cdot \overrightarrow{OC}</math> și <math>\overrightarrow{RT} = b \cdot \overrightarrow{OA} </math>. Rezultă <math>\overrightarrow{MN} = a \cdot \overrightarrow{OA}+ b \cdot \overrightarrow{OB} </math> și <math>\overrightarrow{NT} = a \cdot \overrightarrow{OC} + b \cdot \overrightarrow{OA} </math>. | ||
Dacă <math>a \cdot b > 0,</math> atunci <math>m(\angle MSN) = m(\angle TRN) = 60^\circ</math> , iar daca <math>a \cdot b < 0,</math> atunci <math>m(\angle MSN) = m(\angle TRN) = 120^\circ</math>. Cum <math>MS = RT = |b|,</math> iar <math> SN = NR = |a|,</math> rezultă că triunghiurile <math>MNS</math> și <math>TNR</math> sunt congruente, deci <math>TN = MN </math> și <math>m(\angle MSN) = m(\angle TRN). </math>. | Dacă <math>a \cdot b > 0,</math> atunci <math>m(\angle MSN) = m(\angle TRN) = 60^\circ</math> , iar daca <math>a \cdot b < 0,</math> atunci <math>m(\angle MSN) = m(\angle TRN) = 120^\circ</math>. Cum <math>MS = RT = |b|,</math> iar <math> SN = NR = |a|,</math> rezultă că triunghiurile <math>MNS</math> și <math>TNR</math> sunt congruente, deci <math>TN = MN </math> și <math>m(\angle MSN) = m(\angle TRN). </math>. | ||
Revision as of 15:21, 21 January 2024
28260 (Dana Heuberger)
Enunț
Fie triunghiul echilateral 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 ABC} înscris în cercul de centru 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 O} și rază 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} . Considerăm mulțimea 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 \mathcal{M}} a 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 X} din plan cu proprietatea 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 \overrightarrow{OX} = k \cdot \overrightarrow{OA} + m \cdot \overrightarrow{OB} + n \cdot \overrightarrow{OC}} , unde 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, m, n \in N^*} . Arătați că oricare ar fi punctele distincte 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, N, P \in \mathcal{M} } există 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 Q\in\mathcal{M}} astfel încât vectorii 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 \overrightarrow{MN}} , 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 \overrightarrow{PQ} } 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 \overrightarrow{NM}+} 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 \overrightarrow{QP}} să formeze un triunghi echilateral.
Soluție
Formăm în plan o rețea de triunghiuri echilaterale ale căror vârfuri se află pe drepte paralele echidistante, având direcțiile dreptelor 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 OA} , 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 OB} 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 OC} , distanța ditre două drepte consecutive fiind de 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 \sqrt{3}} . Cum 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 \overrightarrow{OC}} = 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 -\overrightarrow{OA} - \overrightarrow{OB}} obținem 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 X\in \mathcal{M}} dacă și numai dacă există 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, q\in \mathbb{Z}} , astfel încât 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 \overrightarrow{OX} = p\cdot\overrightarrow{OA} + q\cdot\overrightarrow{OB} } .
Analog, coordonatele lui 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 \overrightarrow{OX}} în baza 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 (\overrightarrow{OB},\,\overrightarrow{OC})} , precum și cele din baza 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 (\overrightarrow{OA},\,\overrightarrow{OC})} sunt întregi. De aici rezultă ușor 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 \mathcal{M}} este mulțimea tuturor vârfurilor rețelei.
Alegem punctele 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, N, P \in \mathcal{M} } . Dacă vectorul 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 \overrightarrow{MN}} este paralel cu unul dintre vectorii 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 \overrightarrow{OA}} , 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 \overrightarrow{OB}} , 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 \overrightarrow{OC}} , atunci problema este evidentă. 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 \overrightarrow{MN}} nu este paralel cu niciunul dintre vectorii 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 \overrightarrow{OA}} , 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 \overrightarrow{OB}} , 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 \overrightarrow{OC}} , atunci 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 a, b\in \mathbb{Z}^*} coordonatele vectorului 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 \overrightarrow{MN}} în baza 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 (\overrightarrow{OA}, \overrightarrow{OB})} și punctele 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 S, T, R} astfel încât 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 \overrightarrow{SN} = a \cdot \overrightarrow{OA}} , 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 \overrightarrow{MS} = b \cdot \overrightarrow{OB}} , 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 \overrightarrow{NR} = a \cdot \overrightarrow{OC}} 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 \overrightarrow{RT} = b \cdot \overrightarrow{OA} } . Rezultă 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 \overrightarrow{MN} = a \cdot \overrightarrow{OA}+ b \cdot \overrightarrow{OB} } 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 \overrightarrow{NT} = a \cdot \overrightarrow{OC} + b \cdot \overrightarrow{OA} } .
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 a \cdot b > 0,} 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 m(\angle MSN) = m(\angle TRN) = 60^\circ} , iar daca 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 \cdot b < 0,} 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 m(\angle MSN) = m(\angle TRN) = 120^\circ} . Cum 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 MS = RT = |b|,} iar 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 SN = NR = |a|,} rezultă că triunghiurile 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 MNS} 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 TNR} sunt congruente, 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 TN = MN } 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(\angle MSN) = m(\angle TRN). } .
Întrucât 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(\angle SNR) = 60^\circ} , obținem 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(\angle MNT) = 60^\circ} , deci 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 MNT} este echilateral. Este suficient să alegem punctul 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 Q} astfel încât 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 \overrightarrow{PQ} =} 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 \overrightarrow{NT}} și problema este rezolvată.
Remarcă: De fapt, 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 TNR} este imaginea 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 MNS} prin rotația de centru 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} și unghi de 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 60^/circ} .