Question

设M,N,P是△ABC边上的点,且BM=1/3BC,CN=1/3CA,AP=1/3AB,若A→B=a,A→C=b,用a,

Answer 1

若A→B=a,A→C=b,有

AP=1/3*AB=a/3,

CN=1/3*CA=-b/3.

而,AB+BC=AC,

BC=AC-AB=b-a.

BM=1/3*BC=(b-a)/3,

而,BC=BM+MC,

MC=BC-BM=(b-a)-(b-a)/3=2(b-a)/3.

而,MN=MC+CN=2(b-a)/3-b/3=(b-2a)/3.

而,AC=AN+NC,

AN=AC-NC=b-b/3=2b/3,

而,AN=AP+PN,

PN=AN-AP=2b/3-a/3=(2b-a)/3,

则NP=-(2b-a)/3=(a-2b)/3.

而,AB=AP+PB,

PB=AB-AP=a-a/3=2a/3.

而,PM=PB+BM=2a/3+(b-a)/3=(a+b)/3.