Found problems: 4
2024 Israel National Olympiad (Gillis), P6
Quadrilateral $ABCD$ is inscribed in a circle. Let $\omega_A$, $\omega_B$, $\omega_C$, $\omega_D$ be the incircles of triangles $DAB$, $ABC$, $BCD$, $CDA$ respectively. The common external common tangent of $\omega_A$, $\omega_B$, different from line $AB$, meets the external common tangent of $\omega_A$, $\omega_D$, different from $AD$, at point $A'$. Similarly, the external common tangent of $\omega_B$, $\omega_C$ different from $BC$ meets the external common tangent of $\omega_C$, $\omega_D$ different from $CD$ at $C'$.
Prove that $AA'\parallel CC'$.
1951 Poland - Second Round, 1
In a right triangle $ ABC $, the altitude $ CD $ is drawn from the vertex of the right angle $ C $ and a circle is inscribed in each of the triangles $ ABC $, $ ACD $ and $ BCD $. Prove that the sum of the radii of these circles equals the height $ CD $.
2000 Czech And Slovak Olympiad IIIA, 2
Let be given an isosceles triangle $ABC$ with the base $AB$. A point $P$ is chosen on the altitude $CD$ so that the incircles of $ABP$ and $PECF$ are congruent, where $E$ and $F$ are the intersections of $AP$ and $BP$ with the opposite sides of the triangle, respectively. Prove that the incircles of triangles $ADP$ and $BCP$ are also congruent.
2021/2022 Tournament of Towns, P3
The intersection of two triangles is a hexagon. If this hexagon is removed, six small triangles remain. These six triangles have the same in-radii. Prove the in-radii of the original two triangles are also equal.
Spoiler:
This is one of the highlights of TT. Also SA3