3AP+4BP+5CP=0
即:3AP+4(AP-AB)+5(AP-AC)
=12AP-4AB-5AC=0
即:AP=4AB/12+5AC/12=(4a+5b)/12
令:AP=kPD,则:PD=(4AB+5AC)/(12k)
令:AD=xAB+yAC,则:x+y=1
则:PD=AD-AP=(x-4/12)AB+(y-5/12)AC
故:x-1/3=1/(3k),y-5/12=5/(12k)
即:(x-1/3)/(y-5/12)=4/5
即:5x=4y,即:4y/5+y=1
即:y=5/9,x=4/9
故:AD=4AB/9+5AC/9=(4a+5b)/9