Fie triunghiul \( ABC \) si consideram sirul de puncte definit dupa cum urmeaza:
\( C_0=C, \) \( C_{n+1} \) este centrul cercului inscris in triunghiul \( ABC_n \).
Calculati \( \lim_{n\to \infty}C_n \).
Jarnik 2009, Problema 1
Limita unui sir de puncte in plan
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Beniamin Bogosel
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Limita unui sir de puncte in plan
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Beniamin Bogosel
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Folosind faptul ca centrul cercului inscris are formula vectoriala \( I=\frac{aA+bB+cC}{a+b+c} \), obtinem trei siruri recurente \( a_n,b_n,c_n \) astfel incat \( C_n=a_nA+b_nB+c_nC \). Recurentele se obtin din formula de mai sus.
Rezulta ca \( a_n\to \frac{a}{a+b},b_n \to \frac{b}{a+b}, c_n \to 0 \).
Prin urmare, limita cautata este piciorul bisectoarei din \( C \) a triunghiului.
Ar fi interesanta o solutie cu transformari geometrice.
Rezulta ca \( a_n\to \frac{a}{a+b},b_n \to \frac{b}{a+b}, c_n \to 0 \).
Prin urmare, limita cautata este piciorul bisectoarei din \( C \) a triunghiului.
Ar fi interesanta o solutie cu transformari geometrice.
Yesterday is history,
Tomorow is a mistery,
But today is a gift.
That's why it's called present.
Blog
Tomorow is a mistery,
But today is a gift.
That's why it's called present.
Blog