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lines UD, VE, WF , OI are concurrent.

Source: Indian Postal Coaching 2009 set 1 p3

May 26, 2020
geometryConcyclicCircumcenteraltitudesincenter

Problem Statement

Let ABCABC be a triangle with circumcentre OO and incentre II such that OO is different from II. Let AK,BL,CMAK, BL, CM be the altitudes of ABCABC, let U,V,WU, V , W be the mid-points of AK,BL,CMAK, BL, CM respectively. Let D,E,FD, E, F be the points at which the in-circle of ABCABC respectively touches the sides BC,CA,ABBC, CA, AB. Prove that the lines UD,VE,WFUD, VE, WF and OIOI are concurrent.
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