梯形ABCD中,AD平行与BC,M为CD中点,且AM、BM分别平分∠DAB、∠ABC,求证;AB=AD+BC

Question

梯形ABCD中,AD平行与BC,M为CD中点,且AM、BM分别平分∠DAB、∠ABC,求证;AB=AD+BC

Answer 1

过点M作MN//AD交AB于N,则MN为梯形ABCD的中位线=>MN//=1/2(AD+BC) =>2MN=AD+BC--(1)
AD//BC =>∠DAB+∠ABC=180°=>1/2∠DAB+1/2∠ABC=90° =>∠AMB=90°=>AB=2MN---(2)
(1)(2)=>AB=AD+BC