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radical axis of two circles bisects segment DE

Source: 2018 Saudi Arabia BMO TST II p1

July 25, 2020

radical axisgeometrybisects segment

Problem Statement

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

\angle BPM, \angle MNP

and

J

is the intersection of angle bisectors of

\angle CN M, \angle MPN

. Denote

(\omega_1)

as the circle of center

I

and tangent to

MP

at

D

,

(\omega_2)

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

(\omega_1), (\omega_2)

bisects the segment

DE

.

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