В четырехугольнике АВСМ

lt;A+lt;B+lt;C+lt;M=360
lt;M=140
ACM:
AM=CM. lt;CAM=lt;ACM=(180-140)/2, lt;ACM=lt;CAM=20
ACB: lt;BAC=60-20, lt;BAC=40. =gt;
ACB -равнобедренный. АС=ВС. lt;ACB=120-20. lt;ACB=100
по теореме косинусов:
АВ=АС+ВС-2*АВ*ВС*coslt;ACB. AC=BC=x
10=x+x-2*x*x*cos100
100=2x-2x *cos100, 100=x(2-2cos100)
x=100/(2-2cos100). x=50/(1-cos100)
BC=5(2/(1-cos100))

ACM по аксиоме косинусов:
АС=АМ+СМ-2*АМ*СМ*cos140. AM=CM=y
AC=y+y-2y*y* cos140
AC=2y-2ycos140. AC=y(2-2cos140). 
y=AC/(2-2cos140).  y=AC/(2(1-cos140))
y=(50/(1-cos100))/(2(1-cos140)
y=25/ [ (1-cos100)*(1-cos140)]
y=5/[ (1-cos100)*(1-cos140)], CM=5/ [(1-cos100)*(1-cos140)]

BC+CM=5(2/(1-cos100))+5/[ (1-cos100)*(1-cos140)]