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Lots of concurencies

Source: VNTST 2020

June 27, 2020
geometrycircumcircle

Problem Statement

In acute △ABC\triangle ABC, OO is the circumcenter, II is the incenter. The incircle touches BC,CA,ABBC,CA,AB at D,E,FD,E,F. And the points K,M,NK,M,N are the midpoints of BC,CA,ABBC,CA,AB respectively.
a) Prove that the lines passing through D,E,FD,E,F in parallel with IK,IM,INIK,IM,IN respectively are concurrent.
b) Points T,P,QT,P,Q are the middle points of the major arc BC,CA,ABBC,CA,AB on ⊙ABC\odot ABC. Prove that the lines passing through D,E,FD,E,F in parallel with IT,IP,IQIT,IP,IQ respectively are concurrent.
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