Olympiad problems

1972 Vietnam National Olympiad, 3

$ABC$ is a triangle. $U$ is a point on the line $BC$. $I$ is the midpoint of $BC$. The line through $C$ parallel to $AI$ meets the line $AU$ at $E$. The line through $E$ parallel to $BC$ meets the line $AB$ at $F$. The line through $E$ parallel to $AB$ meets the line $BC$ at $H$. The line through $H$ parallel to $AU$ meets the line $AB$ at $K$. The lines $HK$ and $FG$ meet at $T. V$ is the point on the line $AU$ such that $A$ is the midpoint of $UV$. Show that $V, T$ and $I$ are collinear.

2010 Contests, 1

Let $ ABC$ be a triangle with circum-circle $ \Gamma$. Let $ M$ be a point in the interior of triangle $ ABC$ which is also on the bisector of $ \angle A$. Let $ AM, BM, CM$ meet $ \Gamma$ in $ A_{1}, B_{1}, C_{1}$ respectively. Suppose $ P$ is the point of intersection of $ A_{1}C_{1}$ with $ AB$; and $ Q$ is the point of intersection of $ A_{1}B_{1}$ with $ AC$. Prove that $ PQ$ is parallel to $ BC$.

cyclic quadrilateral geometry angle bisector projective geometry parallel lines angle chasing cyclic quadrilaterals radical axis

2022 European Mathematical Cup, 3

Let $ABC$ be an acute-angled triangle with $AC > BC$, with incircle $\tau$ centered at $I$ which touches $BC$ and $AC$ at points $D$ and $E$, respectively. The point $M$ on $\tau$ is such that $BM \parallel DE$ and $M$ and $B$ lie on the same halfplane with respect to the angle bisector of $\angle ACB$. Let $F$ and $H$ be the intersections of $\tau$ with $BM$ and $CM$ different from $M$, respectively. Let $J$ be a point on the line $AC$ such that $JM \parallel EH$. Let $K$ be the intersection of $JF$ and $\tau$ different from $F$. Prove that $ME \parallel KH$.

geometry angle bisector incircle parallel lines angle chasing homothety law of cosines power of a point similar triangles