If the black triangle below is isosceles, and the vertices of the green
triangle are the midpoints of the sides of the black triangle, that the
green triangle is also isosceled:



Since the black triangle is isosceles we can let it be triangle PQR,
where the vertices are P(0,b), Q(-a,0), and R(a,0), where a and b are
both positive numbers:

 

Now we will let S, T. and U be the midpoints respectively of
PQ, PR and QR.

We will use the midpoint formula to find the coordinates of S,T, and U:

For S:
Midpoint = 
Midpoint of PQ = S
Midpoint of PQ = S

For T:
Midpoint = 
Midpoint of PR = T
Midpoint of PR = T

For U:
Midpoint = 
Midpoint of QR = U
Midpoint of QR = U(0,0)   



Now all we need is to show that US = UT.

We will use the distance tance formula to show that:

 d = 
SU = 
SU = 
SU = 
SU = 
SU = 
SU = 

 d = 
TU = 
TU = 
TU = 
TU = 
TU = 
TU = 

So SU = TU and therefore triangle STU is isosceles.

Edwin