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common ext.tangents of MDP and MPE meet on midline of triangle ABC

Source: Sharygin Finals 2022 10.5

October 26, 2022

geometryconcurrencycommon tangents

Problem Statement

Let

AB

and

AC

be the tangents from a point

A

to a circle

\Omega

. Let

M

be the midpoint of

BC

and

P

be an arbitrary point on this segment. A line

AP

meets

\Omega

at points

D

and

E

. Prove that the common external tangents to circles

MDP

and

MPE

meet on the midline of triangle

ABC