Olympiad problems

2014 Brazil National Olympiad, 6

Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. Circle $\omega_A$ is externally tangent to $\omega$ and tangent to sides $AB$ and $AC$ at $A_1$ and $A_2$, respectively. Let $r_A$ be the line $A_1A_2$. Define $r_B$ and $r_C$ in a similar fashion. Lines $r_A$, $r_B$ and $r_C$ determine a triangle $XYZ$. Prove that the incenter of $XYZ$, the circumcenter of $XYZ$ and $I$ are collinear.

2023-24 IOQM India, 5

In a triangle $A B C$, let $E$ be the midpoint of $A C$ and $F$ be the midpoint of $A B$. The medians $B E$ and $C F$ intersect at $G$. Let $Y$ and $Z$ be the midpoints of $B E$ and $C F$ respectively. If the area of triangle $A B C$ is 480 , find the area of triangle $G Y Z$.

area barycentric geometry area of a triangle barycentric coordinates

2014 PUMaC Individual Finals A, 1

Let $\gamma$ be the incircle of $\triangle ABC$ (i.e. the circle inscribed in $\triangle ABC$) for which $AB+AC=3BC$. Let the point where $AC$ is tangent to $\gamma$ be $D$. Let the incenter of $I$. Let the intersection of the circumcircle of $\triangle BCI$ with $\gamma$ that is closer to $B$ be $P$. Show that $PID$ is collinear.

geometric transformation incenter barycentric circumcircle geometry ring theory analytic geometry Ptolemy's theorem barycentric coordinates