Inegalitate geometrica, lista scurta 2004
Posted: Mon Feb 08, 2010 6:22 pm
Fie \( A^{\prime},B^{\prime},C^{\prime} \) respectiv, puncte pe laturile \( (BC),(AC),(AB) \) ale unui triunghi \( ABC \) şi numărul \( k\geq 1 \) astfel ca:
\( \frac{1}{k}\leq\frac{AC^{\prime}}{BC^{\prime}}\leq K, \ \ \ \ \frac{1}{k}\leq\frac{BA^{\prime}}{CA^{\prime}}\leq K, \ \ \ \ \frac{1}{k}\leq\frac{CB^{\prime}}{AB^{\prime}}\leq K \). Demonstrati ca: \( \frac{\max(A^{\prime}B^{\prime},A^{\prime}C^{\prime},B^{\prime}C^{\prime})}{\max(AB,AC,BC)}\leq\frac{k}{k+1} \).
Dan Ismailescu
\( \frac{1}{k}\leq\frac{AC^{\prime}}{BC^{\prime}}\leq K, \ \ \ \ \frac{1}{k}\leq\frac{BA^{\prime}}{CA^{\prime}}\leq K, \ \ \ \ \frac{1}{k}\leq\frac{CB^{\prime}}{AB^{\prime}}\leq K \). Demonstrati ca: \( \frac{\max(A^{\prime}B^{\prime},A^{\prime}C^{\prime},B^{\prime}C^{\prime})}{\max(AB,AC,BC)}\leq\frac{k}{k+1} \).
Dan Ismailescu