延长PB至B',使PB'=2PB,延长PC至PC',是PC'=3PC,则
向量PA+ 2PB+ 3PC=PA+PB'+PC'=0,
∴P是△AB'C'的重心,
∴S△PAB'=S△PB'C'=S△PC'A,记为S,
S△APB=S/2,S△APC=S/3,S△BPC=S/6,
∴S△ABC=S/2+S/3+S/6=S,
∴S△APC:S△ABC=1:3.
延长PB至B',使PB'=2PB,延长PC至PC',是PC'=3PC,则
向量PA+ 2PB+ 3PC=PA+PB'+PC'=0,
∴P是△AB'C'的重心,
∴S△PAB'=S△PB'C'=S△PC'A,记为S,
S△APB=S/2,S△APC=S/3,S△BPC=S/6,
∴S△ABC=S/2+S/3+S/6=S,
∴S△APC:S△ABC=1:3.