Question 876826
The perimeter of the original triangle is 3*18=54cm.  The triangle formed by the first midpoint connection has sides of 9cm.  Its perimeter is 3*9cm = 27cm.  The third triangle has sides of 4.5cm.  Its perimeter is 3*4.5cm = 13.5cm.<P>
The total perimeter is 3*(18+9+4.5+...)<P>

18+9+4.5... is a geometric series 18/(2^n) where n is 0 to infinity.<P>
The sum of an infinite geometric series xr^0+xr^1+xr^2+...xr^n+... converges to a certain value which is x/(1-r).  In this case x=18 and r=1/2.  So the sum is 18/(1/2) = 18*2 =36.<P>
The total perimeter is 3*36 = 108 cm.