如图,O为三角形ABC的外心,AB=4,AC=2,

Question

如图,O为三角形ABC的外心,AB=4,AC=2,

Answer 1

作OD⊥AB,OE⊥AC,与AB、AC分别交于D、E.
因为O是三角形ABC的外心,
所以OD、OE必垂直平分AB、AC.
向量AM·向量AO
=1/2(AB+AC)·AO
=1/2AB·AO+1/2AC·AO
=1/2·|AB|·(|AO|·cos∠DAO)+1/2·|AC|·(|AO|·cos∠EAO)
=1/2·|AB|·|AD|+1/2·|AC|·|AE|
=1/4·|AB|·|AB|+1/4·|AC|·|AC|
=1/4·4·4+1/4·2·2
=5