Question

如图所示，在△ABC中，点O是BC的中点。过点O的直线分别角直线AB,AC于不同的两点M,N,若 向量AB=m向量AM，向

在这个地址http://wenwen.soso.com/z/q209355066.htm?w=%C8%E7%CD%BC+%D4%DA%C8%FD%BD%C7%D0%CEABC%D6%D0+%B5%E3O%CA%C7BC%D6%D0%B5%E3+%B9%FD%B5%E3O%B5%C4%D6%B1%CF%DF%B7%D6%B1%F0%BD%BB%D6%B1%CF%DFAB%A3%ACAC%D3%DA%B2%BB%CD%AC%B5%C4%C1%BD%B5%E3M%2CN%A1%A3%C8%F4%CF%F2%C1%BFAB%3Dm%CF%F2%C1%BFAM%A3%AC%CF%F2%C1%BFAC%3Dn%CF%F2%C1%BFAN%A3%AC%D4%F2m%2Bn%B5%C4%D6%B5%CE%AA&spi=1&sr=3&w8=%E5%A6%82%E5%9B%BE+%E5%9C%A8%E4%B8%89%E8%A7%92%E5%BD%A2ABC%E4%B8%AD+%E7%82%B9O%E6%98%AFBC%E4%B8%AD%E7%82%B9+%E8%BF%87%E7%82%B9O%E7%9A%84%E7%9B%B4%E7%BA%BF%E5%88%86%E5%88%AB%E4%BA%A4%E7%9B%B4%E7%BA%BFAB%EF%BC%8CAC%E4%BA%8E%E4%B8%8D%E5%90%8C%E7%9A%84%E4%B8%A4%E7%82%B9M%2CN%E3%80%82%E8%8B%A5%E5%90%91%E9%87%8FAB%3Dm%E5%90%91%E9%87%8FAM%EF%BC%8C%E5%90%91%E9%87%8FAC%3Dn%E5%90%91%E9%87%8FAN%EF%BC%8C%E5%88%99m%2Bn%E7%9A%84%E5%80%BC%E4%B8%BA&qf=20&rn=102&qs=4&ch=w.search.3第二个答案中为什么
向量AB/向量AM=向量PB/向量PO 向量AC/向量AN=向量PC/向量PO

Answer 1

因为三角形APB与三角行MOB是相似三角形，所以BO/PB=BM/AB(相似比)，AM/AB=(AB+BM)/AB=1+BM/AB,PO/PB=(PB+BO)/PB=1+BO/PB,所以向量AB/向量AM=向量PB/向量PO；同理 向量AC/向量AN=向量PC/向量PO