連接OM,ON，將△OMN順時(shí)針旋轉(zhuǎn)60°得到△OM'N',連接OO',N'G

∵∠OMO'=60°，OM=OM'

∴△OMO'為等邊三角形

∴OO'=OM=O'M

∴OM+ON+OG=OO'+O'N'+OG

∴當(dāng)O',N',O,G四點(diǎn)共線時(shí)值最小,即N'G.

過(guò)點(diǎn)G作GF⊥N'M的延長(zhǎng)線于點(diǎn)F

∵∠NMN'=60°，∠NMG=75°,GF⊥N'F

∴∠FMG=∠FGM=45°

∴FM=FG

又∵M(jìn)N=6,MG=4√2

∴MN'=6,FM=FG=4,N'G=√N(yùn)'F2+FG2=2√29

∴點(diǎn)O到△MNG三個(gè)頂點(diǎn)的距離和的最小值是2√29