JBTST III 2007, Problema 1
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Laurian Filip
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JBTST III 2007, Problema 1
Fie ABC un triunghi si punctele M,N,P pe laturile AB,BC,CA respectiv, astfel incat CPMN sa fie paralelogram. Dreptele AN si MP se intersecteaza in punctul R, dreptele BP si MN se intersecteaza in punctul S, iar Q este punctul de intersectie al dreptelor AN si BP. Sa se arate ca S[MRQS]=S[NQP].
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Mateescu Constantin
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Aplicand teorema lui Thales obtinem:
\( \left\|\begin{array}{ccc} \frac{AR}{RN}&=&\frac{AM}{MB}\\\\\\\\\\
\frac{MS}{SN}&=&\frac{SP}{SB}&=&\frac{NC}{NB}\\\\\\\\\\
\frac{AM}{MB}&=&\frac{NC}{NB}\end{array}\right| \ \Longrightarrow\ \frac{AR}{RN}\ =\ \frac{MS}{SN}\ \Longrightarrow^{\mbox{R.T.Th}}\ RS\ \parallel\ AB. \)
Atunci in trapezele \( RPNB \) si \( RSBM \) avem relatiile \( S_{PQN}\ =\ S_{RQB} \) si \( S_{RSB}\ =\ S_{RSM} \), ceea ce conduce la concluzia problemei.
\( \left\|\begin{array}{ccc} \frac{AR}{RN}&=&\frac{AM}{MB}\\\\\\\\\\
\frac{MS}{SN}&=&\frac{SP}{SB}&=&\frac{NC}{NB}\\\\\\\\\\
\frac{AM}{MB}&=&\frac{NC}{NB}\end{array}\right| \ \Longrightarrow\ \frac{AR}{RN}\ =\ \frac{MS}{SN}\ \Longrightarrow^{\mbox{R.T.Th}}\ RS\ \parallel\ AB. \)
Atunci in trapezele \( RPNB \) si \( RSBM \) avem relatiile \( S_{PQN}\ =\ S_{RQB} \) si \( S_{RSB}\ =\ S_{RSM} \), ceea ce conduce la concluzia problemei.