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Found problems: 2

1988 Brazil National Olympiad, 4
Two triangles are circumscribed to a circumference. Show that if a circumference containing five of their vertices exists, then it will contain the sixth vertex too.
circumcircle geometry circles incircle plane geometry porism

Kvant 2022, M2692
In the circle $\Omega$ the hexagon $ABCDEF$ is inscribed. It is known that the point $D{}$ divides the arc $BC$ in half, and the triangles $ABC$ and $DEF$ have a common inscribed circle. The line $BC$ intersects segments $DF$ and $DE$ at points $X$ and $Y$ and the line $EF$ intersects segments $AB$ and $AC$ at points $Z$ and $T$ respectively. Prove that the points $X, Y, T$ and $Z$ lie on the same circle. [i]Proposed by D. Brodsky[/i]
hexagon geometry porism power of a point