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

2009 Iran Team Selection Test, 9
In triangle $ABC$, $D$, $E$ and $F$ are the points of tangency of incircle with the center of $I$ to $BC$, $CA$ and $AB$ respectively. Let $M$ be the foot of the perpendicular from $D$ to $EF$. $P$ is on $DM$ such that $DP = MP$. If $H$ is the orthocenter of $BIC$, prove that $PH$ bisects $ EF$.
graphing lines geometric transformation trigonometry homothety iran lemma geometry slope Menelaus' theorem angle chasing cyclic quadrilaterals radical axis

2002 IMO Shortlist, 7
The incircle $ \Omega$ of the acute-angled triangle $ ABC$ is tangent to its side $ BC$ at a point $ K$. Let $ AD$ be an altitude of triangle $ ABC$, and let $ M$ be the midpoint of the segment $ AD$. If $ N$ is the common point of the circle $ \Omega$ and the line $ KM$ (distinct from $ K$), then prove that the incircle $ \Omega$ and the circumcircle of triangle $ BCN$ are tangent to each other at the point $ N$.
radical axis incenter iran lemma geometry polars Pythagorean theorem cyclic quadrilaterals geometric inversion homothety power of a point