Question 58
Let $ABCD$ be a cyclic quadrilateral. Suppose that there exists a line $l \parallel BD$, which is tangent to the inscribed circles of triangles $ABC$ and $CDA$. Prove that the line $l$ contains the incenter of one of $\triangle BCD$ and $\triangle DAB$.
(Fedir Yudin)
Editorial note. The source names “triangle $ABCD$” as the first triangle in the conclusion. Four vertices do not define a triangle. This is explicitly restored here to $BCD$, the other triangle cut off by diagonal $BD$ besides $DAB$, consistent with interchanging $A,C$ in the source argument. The hypotheses are unchanged.
(Fedir Yudin)
Editorial note. The source names “triangle $ABCD$” as the first triangle in the conclusion. Four vertices do not define a triangle. This is explicitly restored here to $BCD$, the other triangle cut off by diagonal $BD$ besides $DAB$, consistent with interchanging $A,C$ in the source argument. The hypotheses are unchanged.