Question

o为三角形ABC的外心,AB=4,AC=2,角BAC为钝角,M是边BC的中点,求向量AM与向量AO的点积

Answer 1

过O作ON垂直于AC与N,

因角AOC=2角B,故角AON=角B,角OAC=90-角AON=90-角B,

角BAC=180-B-C,角BAO=BAC-OAC=180-B-C-(90-B)=90-C,

由正弦定理AB/sinC=AC/sinB=2R=2AO(R为外接圆半径),故AO=1/sinB,sinC/sinB=2,

向量AB点乘AO=AB\AOcosBAO=(4/sinB)*cos(90-C)=4sinC/sinB=8,

向量AC点乘AO=AC\AOcosOAC=(2/sinB)*cos(90-B)=2sinB/sinB=2,

因向量AM=(AB+AC)/2,

故向量AM点乘AO=[(AB+AC).AO]/2=(AB.AO+AC.AO)/2=(8+2)/2=5.