Teorema lui Steiner
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Andi Brojbeanu
- Bernoulli
- Posts: 294
- Joined: Sun Mar 22, 2009 6:31 pm
- Location: Targoviste (Dambovita)
Teorema lui Steiner
Se considera triunghiul \( ABC \) si punctele\( A\prim, A^{\prime \prime}\in (BC) \). Daca \( m(\angle{A^\prim AB})=m(\angle{A^{\prime \prime}AC}) \), atunci: \( \frac{A^\prim B}{A^\prim C}\cdot \frac{A^{\prime \prime}B}{A^{\prime \prime}C}=\frac{AB^2}{AC^2} \).
Last edited by Andi Brojbeanu on Thu Jun 18, 2009 2:49 pm, edited 1 time in total.
-
Andi Brojbeanu
- Bernoulli
- Posts: 294
- Joined: Sun Mar 22, 2009 6:31 pm
- Location: Targoviste (Dambovita)
Fie punctele \( B_1=pr_{AA^\prim}B \), \( C_1=pr_{AA^\prim}C \),\( B_2=pr_{AA^{\prim\prim}}B \), \( C_2=pr_{AA^{\prim\prim}}C \).
Din asemanarea triunghiurilor dreptunghice \( A^\prim BB_1 \) si \( A^\prim CC_1 \) rezulta:
\( \frac{A^\prim B}{A^\prim C}=\frac{BB_1}{CC_1} \).
Din asemanarea triunghiurilor dreptunghice \( A^{\prim\prim} BB_2 \) si \( A^{\prim\prim} CC_2 \) rezulta:
\( \frac{A^{\prim\prim}B}{A^{\prim\prim}C}=\frac{BB_2}{CC_2} \).
Inmultind cele doua relatii, obtinem:
\( \frac{A^\prim B}{A^\prim C}\cdot \frac{A^{\prim\prim}B}{A^{\prim\prim}C}=\frac{BB_1}{CC_1}\cdot \frac{BB_2}{CC_2}^{(1)} \).
\( m(\angle{BAB_2})=m(\angle{A^\prim A A^{\prim\prim}})+m(\angle{BAA^\prim})=m(\angle{A^\prim A A^{\prim\prim}})+m(\angle{CAA^{\prim\prim}})=m(\angle{CAC_1})\Rightarrow \bigtriangleup{ABB_2}\sim \bigtriangleup{ACC_1}\Rightarrow \frac{BB_2}{CC_1}=\frac{AB}{AC}^{(2)}. \)
\( m(\angle{BAB_1})=m(\angle{CAC_2})\Rightarrow \bigtriangleup{BAB_1}\sim \bigtriangleup{CAC_2}\Rightarrow \frac{BB_1}{CC_2}=\frac{AB}{AC}^{(3)}. \)
\( (1), (2), (3)\Rightarrow \frac{A^\prim B}{A^\prim C}\cdot \frac{A^{\prime \prime}B}{A^{\prime \prime}C}=\frac{AB^2}{AC^2} \), relatia lui Steiner.
Din asemanarea triunghiurilor dreptunghice \( A^\prim BB_1 \) si \( A^\prim CC_1 \) rezulta:
\( \frac{A^\prim B}{A^\prim C}=\frac{BB_1}{CC_1} \).
Din asemanarea triunghiurilor dreptunghice \( A^{\prim\prim} BB_2 \) si \( A^{\prim\prim} CC_2 \) rezulta:
\( \frac{A^{\prim\prim}B}{A^{\prim\prim}C}=\frac{BB_2}{CC_2} \).
Inmultind cele doua relatii, obtinem:
\( \frac{A^\prim B}{A^\prim C}\cdot \frac{A^{\prim\prim}B}{A^{\prim\prim}C}=\frac{BB_1}{CC_1}\cdot \frac{BB_2}{CC_2}^{(1)} \).
\( m(\angle{BAB_2})=m(\angle{A^\prim A A^{\prim\prim}})+m(\angle{BAA^\prim})=m(\angle{A^\prim A A^{\prim\prim}})+m(\angle{CAA^{\prim\prim}})=m(\angle{CAC_1})\Rightarrow \bigtriangleup{ABB_2}\sim \bigtriangleup{ACC_1}\Rightarrow \frac{BB_2}{CC_1}=\frac{AB}{AC}^{(2)}. \)
\( m(\angle{BAB_1})=m(\angle{CAC_2})\Rightarrow \bigtriangleup{BAB_1}\sim \bigtriangleup{CAC_2}\Rightarrow \frac{BB_1}{CC_2}=\frac{AB}{AC}^{(3)}. \)
\( (1), (2), (3)\Rightarrow \frac{A^\prim B}{A^\prim C}\cdot \frac{A^{\prime \prime}B}{A^{\prime \prime}C}=\frac{AB^2}{AC^2} \), relatia lui Steiner.