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 ...?