This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 2

1982 Bulgaria National Olympiad, Problem 6
Find the locus of centroids of equilateral triangles whose vertices lie on sides of a given square $ABCD$.
triangle geometry centroid locus square triangle centers triangles

2023 Israel TST, P3
Let $ABC$ be an acute-angled triangle with circumcenter $O$ and incenter $I$. The midpoint of arc $BC$ of the circumcircle of $ABC$ not containing $A$ is denoted $S$. Points $E, F$ were chosen on line $OI$ for which $BE$ and $CF$ are both perpendicular to $OI$. Point $X$ was chosen so that $XE\perp AC$ and $XF\perp AB$. Point $Y$ was chosen so that $YE\perp SC$ and $YF\perp SB$. $D$ was chosen on $BC$ so that $DI\perp BC$. Prove that $X$, $Y$, and $D$ are collinear.
incenter circumcircle geometry triangle centers angle chasing cyclic quadrilaterals geometric inversion spiral similarity