MathDB
Orthocentre of DEF is incentre of ABC

Source: Romanian MO 2010 Grade 9

August 6, 2012
geometryincentergeometry proposed

Problem Statement

In a triangle ABCABC denote by D,E,FD,E,F the points where the angle bisectors of ∠CAB,∠ABC,∠BCA\angle CAB,\angle ABC,\angle BCA respectively meet it's circumcircle. a) Prove that the orthocenter of triangle DEFDEF coincides with the incentre of triangle ABCABC. b) Prove that if AD→+BE→+CF→=0\overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF}=0, then the triangle ABCABC is equilateral.
Marin Ionescu
Back to ProblemsView on AoPS