Question 23
Let $k_1$, $k_2$ and $k_3$ be three circles with centers $O_1$, $O_2$ and $O_3$ respectively, such that none of the centers lies inside any of the two other circles. The circles $k_1$ and $k_2$ intersect in $A$ and $P$, $k_1$ and $k_3$ intersect in $C$ and $P$ and $k_2$ and $k_3$ intersect in $B$ and $P$. Let $X$ be a point on $k_1$ such that the intersection of the line $XA$ with the circle $k_2$ is $Y$ and the intersection of the line $XC$ with $k_3$ is $Z$ and so that $Y$ belongs neither inside $k_1$ nor inside $k_3$ and $Z$ belongs neither inside $k_1$ nor inside $k_2$.
a) Prove that the triangles $XYZ$ and $O_1O_2O_3$ are similar.
b) Prove that the area of the triangle $XYZ$ is not greater than four times the area of the triangle $O_1O_2O_3$. Is the maximum attainable?
Macedonian source version, translated in full:
Let $k_1$, $k_2$ and $k_3$ be three circles with centers $O_1$, $O_2$ and $O_3$ respectively, such that none of the centers lies inside any of the two other circles. The circles $k_1$ and $k_2$ intersect in $A$ and $P$, $k_1$ and $k_3$ intersect in $C$ and $P$ and $k_2$ and $k_3$ intersect in $B$ and $P$. Let $X$ be a point on $k_1$ such that the intersection of the line $XA$ with the circle $k_2$ is $Y$ and the intersection of the line $XC$ with $k_3$ is $Z$ and so that $Y$ belongs neither inside $k_1$ nor inside $k_3$ and $Z$ belongs neither inside $k_1$ nor inside $k_2$.
a) Prove that the triangles $XYZ$ and $O_1O_2O_3$ are similar.
b) Prove that the area of the triangle $XYZ$ is not greater than four times the area of the triangle $O_1O_2O_3$. Is the maximum attainable?
a) Prove that the triangles $XYZ$ and $O_1O_2O_3$ are similar.
b) Prove that the area of the triangle $XYZ$ is not greater than four times the area of the triangle $O_1O_2O_3$. Is the maximum attainable?
Macedonian source version, translated in full:
Let $k_1$, $k_2$ and $k_3$ be three circles with centers $O_1$, $O_2$ and $O_3$ respectively, such that none of the centers lies inside any of the two other circles. The circles $k_1$ and $k_2$ intersect in $A$ and $P$, $k_1$ and $k_3$ intersect in $C$ and $P$ and $k_2$ and $k_3$ intersect in $B$ and $P$. Let $X$ be a point on $k_1$ such that the intersection of the line $XA$ with the circle $k_2$ is $Y$ and the intersection of the line $XC$ with $k_3$ is $Z$ and so that $Y$ belongs neither inside $k_1$ nor inside $k_3$ and $Z$ belongs neither inside $k_1$ nor inside $k_2$.
a) Prove that the triangles $XYZ$ and $O_1O_2O_3$ are similar.
b) Prove that the area of the triangle $XYZ$ is not greater than four times the area of the triangle $O_1O_2O_3$. Is the maximum attainable?