AoPS - Problem

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Problem

Source: Second Romanian JBMO TST 2016

Tags: geometry

den_thewhitelion

15.06.2016 09:34

Triangle $\triangle{ABC}$ ,O=circumcenter of (ABC),OA=R,the A-excircle intersect (AB),(BC),(CA) at points F,D,E. If the A-excircle has radius R prove that $OD\perp EF$

l1090107005

15.06.2016 09:55

Let $M$ be midpoint of arc $BC$ not containing $A$ of circumcircle of $\triangle{ABC}$ , $J$ be excenter. Then $DJ=OM=R,DJ\perp BC,OM\perp BC$ , so $ODJM$ is parallelogram. And $AJ\perp EF$ , so $OD\perp EF$ .