Bariz - Problem

Source: 2010 Peru Iberoamerican TST problem 3

Tags: geometry

hectorleo123

11.05.2023 02:23

Let $C_1$ and $C_2$ be two concentric circles with center $O$, in such a way that the radius of $C_1$ is smaller than the radius of $C_2$. Let $P$ be a point other than $O$ that is in the interior of $C_1$, and $L$ a line through $P$ and intersects $C_1$ at $A$ and $B$. Ray $\overrightarrow{OB}$ intersects $C_2$ at $C$. Determine the locus that determines the circumcenter of triangle $ABC$ as $L$ varies.

Let ray $OA$ intersect $C_2$ at $D$, then $ABCD$ is a isosceles trapezoid, which is cyclic, so the circumcenter of this trapezoid is the intersection of the perpendicular bisectors of $BC$ and $AD$. Let the midpoint of $BC$ and $AD$ be $M,N$, then $OM=ON$. Let the circle centered at $O$ passing through $M,N$ be $C_3$, then the circumcenter of $ABCD$ is the intersection of the tangents of $M,N$ to $C_3$. If the homothety that takes $B,A$ to $M,N$ from $O$ takes $P$ to $P'$, then $MN$ passes through $P'$, which is a fixed point. So the circumcenter of $ABCD$ lies on the polar of $P'$ wrt $C_3$, i.e. the locus is a line.