Solution: The vertical line with E'C passing through point B intersects its extension line at point F, the vertical line with CM passing through point D intersects it at point H, and the vertical line with CM passing through point A intersects its extension line at point G. 。

∠∠ACD ' = 60，∠D'CE'=90，

∴∠ACE'=30

∠∠ACB = 90，∠ FCE' = 180。

∴∠BCF=∠FCE'-∠ACB-∠ACE'=60，

BF=sin∠BCF？ BC=5√3

∴S△BCE'= 1/2？ BF？ CE'= 15√3

∠∠ACG+∠BCN = 90，∠BCN+∠CBN=90，

∴∠ACG=∠CBN

AC = BC，

∴Rt△ACG≌Rt△CBN，

∴AG=CN,CG=BN.

Similarly △ CD ′ h △ ce ′ n, d ′ h = cn, CH = NE' ′.

∴M is the midpoint of GH, and cm =1/2 (CG+ch) =1/2be'.

And ∵BF=5√3, ∠ BCF = 60,

∴cf=5,fe′=cf+ce′= 1 1，

∴BE'=√(BF^2+FE'^2)= 14，

∴CM= 1/2BE'=7.

∫S△BCE ' = 1/2？ CN？ Yes,

∴cn=2s△bce′÷be'= 15√3/7

∴MN=CM-CN=7- 15√3/7

Finally finished ...?