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Found problems: 1

2023 Bulgaria EGMO TST, 6
Tags: mixtilinear , brianchon , ceva theorem , radical axis , concurrency , incircle , geometry
Let $ABC$ be a triangle with incircle $\gamma$. The circle through $A$ and $B$ tangent to $\gamma$ touches it at $C_2$ and the common tangent at $C_2$ intersects $AB$ at $C_1$. Define the points $A_1$, $B_1$, $A_2$, $B_2$ analogously. Prove that: a) the points $A_1$, $B_1$, $C_1$ are collinear; b) the lines $AA_2$, $BB_2$, $CC_2$ are concurrent.