circle O1 is tangent to AC, BC(side of triangle ABC) at point D,E.
circle O2 include O1, is tangent to BC, AB(side of triangle ABC) at point E,F
The tangent of O2 at P(DE∩O2,P=E) meets AB at Q.
A line passing through O1(center of O1) and parallel to BO2(O2 is also center of O2) meets BC at G, EQ∩AC=K,KG∩EF=L, EO2 meets circle O2 at N(=E), LO2∩FN=M.
IF N is a middle point of FM, prove that BG=2EG