Problem

Source: 2018 Saudi Arabia BMO TST II p1

Tags: radical axis, geometry, bisects segment



Let ABC be a triangle with M,N,P as midpoints of the segments BC,CA,AB respectively. Suppose that I is the intersection of angle bisectors of BPM,MNP and J is the intersection of angle bisectors of CNM,MPN. Denote (ω1) as the circle of center I and tangent to MP at D, (ω2) as the circle of center J and tangent to MN at E. a) Prove that DE is parallel to BC. b) Prove that the radical axis of two circles (ω1),(ω2) bisects the segment DE.