`#bot`

ta có:

`vec(AM)=vec(AC)+vec(CM)`

`=vec(AC)+1/3 vec(CB)`

`=vec(AC)+1/3(vec(CA)+vec(AB))`

`=vec(AC)-1/3vec(AC)+1/3vec(AB)`

`=2/3vec(AC)+1/3vec(AB)`

;

đặt `PA=x(x>0)`

`vec(PN)=vec(PA)+vec(AN)`

`=x/3 vec(BA)+1/3 vec(AC)`

do `AM bot PN` nên:

`vec(AM)*vec(PN)=0`

`<=>(2/3vec(AC)+1/3vec(AB))*(x/3 vec(BA)+1/3vec(AC))=0`

`<=>-(2x)/9 vec(AC)*vec(AB)+2/9(vec(AC))^2-x/9(vec(AB))^2+1/9vec(AB)*vec(AC)=0`

ta có: 

`{((vec(AC))^2=(vec(AB))^2=AB^2=9),(vec(AB)*vec(AC)=AC*AB*cos (AB,AC)=3*3*cos60^o=9/2):}`

suy ra:

`-(2x)/9 * 9/2 +2/9 *9-x/9 *9+1/9 *9/2=0`

`<=>-x+2-x+1/2=0`

`<=>-2x=-5/2`

`<=>x=5/4` (tm)

`=>PA=5/4`