In an acute triangle segments and are altitudes. Two circles passing through the point anf and tangent to the line at the points and so that lies between and . Prove that lines and intersect on the circumcircle of triangle .
Proposed by Davood Vakili, Iran
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In an acute triangle $ABC$ segments $BE$ and $CF$ are altitudes. Two circles passing through the point $A$ anf $F$ and tangent to the line $BC$ at the points $P$ and $Q$ so that $B$ lies between $C$ and $Q$. Prove that lines $PE$ and $QF$ intersect on the circumcircle of triangle $AEF$.
Proposed by Davood Vakili, Iran
Izvor: Međunarodna matematička olimpijada, shortlist 2008
Školjka 
segments 
and 
are altitudes. Two circles passing through the point 
anf 
and tangent to the line 
at the points 
and 
so that 
lies between 
and 
and 
intersect on the circumcircle of triangle 
.