Se considera un triunghi \( ABC \). Fie \( A_1 \) un punct din plan cu proprietatile:
-dreapta \( BC \) separa punctele \( A \) si \( A_1 \);
-\( A_1 \) este centrul patratului construit pe latura \( [BC] \).
Analog se obtin punctele \( B_1 \) si \( C_1 \). Atunci dreptele \( AA_1, BB_1 \) si \( CC_1 \) sunt concurente. (Punctul de concurenta a celor trei drepte se numeste punctul lui Vecten).
Teorema lui Vecten
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Andi Brojbeanu
- Bernoulli
- Posts: 294
- Joined: Sun Mar 22, 2009 6:31 pm
- Location: Targoviste (Dambovita)
-
Andi Brojbeanu
- Bernoulli
- Posts: 294
- Joined: Sun Mar 22, 2009 6:31 pm
- Location: Targoviste (Dambovita)
Din asemanarea triunghiurilor \( ACB_1 \) si \( ABC_1 \) rezulta \( \frac{AB}{AC}=\frac{AC_1}{AB_1} \) sau \( AB\cdot AB_1=AC\cdot AC_1^{(1)} \).
Deoarece \( m(\angle{BAB_1})=m(\angle{CAC_1})\Rightarrow sin \angle{BAB_1}=sin \angle{CAC_1}^{(2)} \).
\( (1) \) si \( (2)\Rightarrow AB\cdot AB_1\cdot sin \angle{BAB_1}=AC\cdot AC_1\cdot sin \angle{CAC_1}\Rightarrow \sigma[ABB_1]=\sigma[ACC_1]^{(3)} \).
In mod asemanator se obtine \( \sigma[BCC_1]=\sigma [BAA_1] \) si \( \sigma[CAA_1]=\sigma[CBB_1]^{(4)} \).
Fie punctele \( \{A^\prim\}=BC\cap AA_1 \), \( \{B^\prim\}=CA\cap BB_1 \), \( \{C^\prim\}=AB\cap CC_1 \).
Se observa ca:
\( \frac{A^\prim B}{A^\prim C}=\frac{\sigma[BAA_1]}{\sigma[CAA_1]}^{(5)} \)
\( \frac{B^\prim C}{B^\prim A}=\frac{\sigma[CBB_1]}{\sigma[ABB_1]}^{(6)} \)
\( \frac{C^\prim A}{C^\prim B}=\frac{\sigma[ACC_1]}{\sigma[BCC_1]}^{(7)} \)
\( (3), (4), (5), (6), (7)\Rightarrow \frac{A^\prim B}{A\prim C}\cdot \frac{B^\prim C}{B^\prim A}\cdot \frac{C^\prim A}{C^\prim B}=1 \), din care, conform Reciprocei teoremei lui Ceva rezulta ca dreptele \( AA_1, BB_1,CC_1 \) sunt concurente (Teorema lui Vecten).
Deoarece \( m(\angle{BAB_1})=m(\angle{CAC_1})\Rightarrow sin \angle{BAB_1}=sin \angle{CAC_1}^{(2)} \).
\( (1) \) si \( (2)\Rightarrow AB\cdot AB_1\cdot sin \angle{BAB_1}=AC\cdot AC_1\cdot sin \angle{CAC_1}\Rightarrow \sigma[ABB_1]=\sigma[ACC_1]^{(3)} \).
In mod asemanator se obtine \( \sigma[BCC_1]=\sigma [BAA_1] \) si \( \sigma[CAA_1]=\sigma[CBB_1]^{(4)} \).
Fie punctele \( \{A^\prim\}=BC\cap AA_1 \), \( \{B^\prim\}=CA\cap BB_1 \), \( \{C^\prim\}=AB\cap CC_1 \).
Se observa ca:
\( \frac{A^\prim B}{A^\prim C}=\frac{\sigma[BAA_1]}{\sigma[CAA_1]}^{(5)} \)
\( \frac{B^\prim C}{B^\prim A}=\frac{\sigma[CBB_1]}{\sigma[ABB_1]}^{(6)} \)
\( \frac{C^\prim A}{C^\prim B}=\frac{\sigma[ACC_1]}{\sigma[BCC_1]}^{(7)} \)
\( (3), (4), (5), (6), (7)\Rightarrow \frac{A^\prim B}{A\prim C}\cdot \frac{B^\prim C}{B^\prim A}\cdot \frac{C^\prim A}{C^\prim B}=1 \), din care, conform Reciprocei teoremei lui Ceva rezulta ca dreptele \( AA_1, BB_1,CC_1 \) sunt concurente (Teorema lui Vecten).