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jonathanwoenardi

Hitunglah AG : GB

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Misalkan $\omega_A,\omega_B,\omega_C$ adalah tiga lingkaran identik berpusat di $A,B,C$ secara berurutan sehinggga $\omega_A$ melalui $B$ dan $C$, $\omega_B$ melalui $C$ dan $A$, $\omega_C$ melalui $A$ dan $B$. $\omega_A$ memotong $\omega_B$ di $C$ dan $F$, sementara $\omega_C$ di $B$ dan $E$. $CF$ memotong $\omega_C$ di $D$ dan lingkaran berpusat di $E$ melalui $D$ memotong $AB$ di $G$. Hitung nilai perbandingan $AG : GB$.

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Misalkan jari2 lingkaranny $r$. Jelas $\overline{EC}=\overline{CD}=r$, jadi $\overline{DE} =r\sqrt{2} =\overline{EG}$.

Namakan titik $X, Y, Z$ di sepanjang garis $EC$ shg $\overline{GX}, \overline{BY} \perp \overline{EC}$ dan $Z$ perpotongan lingk $\omega_{B}, \omega_{C}$.

Pandang segitiga siku2 $EGX$ dg $\overline{GX}=\frac{x\sqrt{3}}{2}$. Didapat $\overline{EX}=\frac{x\sqrt{5}}{2}$. Sdgkn pada segitiga $YBZ$, $\overline{YZ} = \frac{ZB}{2}= \frac{r}{2}$. Jd $\overline{GB}=\overline{XY}=\overline{EZ}-\overline{EX}-\overline{YZ}=\frac{(3-\sqrt{5})r}{2}$.

=> $AG:GB= (\sqrt{5}-1):(3-\sqrt{5})$. CMIIW.

 

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