Question 84
Let $P$ be an inner point of an acute triangle $ABC$, $E$ and $F$ be the projections of $P$ onto lines $AC$ and $AB$, respectively, and the lines $BP$ and $CP$ intersect the circumcircle of the triangle $ABC$ at points $B_1$ and $C_1$ ($B_1 \neq B$, $C_1 \neq C$), respectively.
Let $R$ and $r$ denote the radii of the circumcircle and incircle of the triangle $ABC$. Prove that $\frac{EF}{B_1C_1} \geq \frac{r}{R}$. Furthermore, determine all positions of $P$ for which the equality holds.
Let $R$ and $r$ denote the radii of the circumcircle and incircle of the triangle $ABC$. Prove that $\frac{EF}{B_1C_1} \geq \frac{r}{R}$. Furthermore, determine all positions of $P$ for which the equality holds.