Two circles $\omega_1$ and $\omega_2$ intersecting at points $X{}$ and $Y{}$ are inside the circle $\Omega$ and touch it at points $A{}$ and $B{}$, respectively; the segments $AB$ and $XY$ intersect. The line $AB$ intersects the circles $\omega_1$ and $\omega_2$ again at points $C{}$ and $D{}$, respectively. The circle inscribed in the curved triangle $CDX$ touches the side $CD$ at the point $Z{}$. Prove that $XZ$ is a bisector of $\angle AXB{}$.
Problem
Source: Russian TST 2016, Day 8 P3 (Groups A & B)
Tags: geometry, circles
khina
19.04.2023 21:15
Let the inscribed circle be $\omega$. Let it be tangent to $\omega_1, \omega_2$ at $P, Q$, and let the exsimilicenter of $\omega_1, \omega_2$ be $S$. Monge on $\omega, \omega_1, \omega_2$ and $\Omega, \omega_1, \omega_2$ gives us that $A, B, S$ and $P, Q, S$ are collinear. Then inverting about $S$ with radius $SX$ clearly fixes all four relevant circles, and in particular it fixes $Z$. So $SX^2 = SZ^2 = SA \times SB$, which is well-known to imply that $XZ$ bisects $\angle{AXB}$.
BTW - are problems ordered by difficulty? How do the days work? @below thanks for the information (and for posting these problems)! I don't think the problem is far too easy (I think it is a little tricky, about a problem 2), was just wondering.
oVlad
19.04.2023 21:37
khina wrote:
Let the inscribed circle be $\omega$. Let it be tangent to $\omega_1, \omega_2$ at $P, Q$, and let the exsimilicenter of $\omega_1, \omega_2$ be $S$. Monge on $\omega, \omega_1, \omega_2$ and $\Omega, \omega_1, \omega_2$ gives us that $A, B, S$ and $P, Q, S$ are collinear. Then inverting about $S$ with radius $SX$ clearly fixes all four relevant circles, and in particular it fixes $Z$. So $SX^2 = SZ^2 = SA \times SB$, which is well-known to imply that $XZ$ bisects $\angle{AXB}$.
BTW - are problems ordered by difficulty? How do the days work? They should be ordered in difficulty. Also, the days are complicated, from what I heard some tests are used as training for some groups, other tests are used as selection for the next year etc. I really am not sure, just enjoy the problems. Edit: when it comes to group tests, I think the "real ones" are the NG tests.