07/04/2026

$A$ is a point on $\odot{O_1}$ and $O_1A=2r$ is the diameter of $\odot{O_2}$, $BC \perp O_1A$, $B$ is on $\odot{O_2}$, $C$ is on $\odot{O_1}$, find the maximum value of $BC$.

Solve:

\(\begin{multline} \shoveleft \text{Make }O_2O_3 \perp O_1A, O_1O_3=3r \text{ and the radius of } \odot{O_3}=r\\ \shoveleft \text{Let } \odot{O_1}\cap \odot{O_3}=C_m, B_mC_m \parallel BC, B_mC_m \cap \odot{O_2}=B_m\\ \shoveleft \text{Make }O_1O_1'\parallel BC \parallel AA' \text{ and }O_1', A' \text{ are on }\odot{O_3}\\ \shoveleft \text{Extend }BC \text{ to }B' \text{ such that }BC \cap \odot{O_3}=B'\\ \shoveleft \implies O_1O_1'=BB'=AA'=B_mC_m=O_2O_3=\sqrt{r^2+(3r)^2}=\sqrt{10}r\\ \shoveleft \implies B_mC_m \text{ is the maximum value of }BC=\bbox[5px, border: 1px solid black]{\sqrt{10}r}\\ \end{multline}\)

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