Starting diagram...

 A = (0, 0)
 B = (1, 0)
 C = (1, 1)
 D = (0, 1)
 E = (1/2, 0)
 F = (0, 1/2)
 G = (1/2, 1/2)
 H = (1/4, 1/4)
 I = (3/4, 1/4)
 
Second diagram...

 A = (0, 0)
 B = (1, 0)
 C = (1, 1)
 D = (0, 1)
 E = (1/2, 0)
 F = (0, 1/2)
 G = (1/2, 1/2)
 H = (1/4 + x, 1/4)
 I = (3/4, 1/4)
 
Third diagram...

 A = (0, 0)
 B = (1, 0)
 C = (1, 1)
 D = (0, 1)
 E = (1/2, 0)
 F = (0, 1/2)
 G = (1/2, 1/2)
 H = (1/4 + x, 1/4)
 I = (3/4 - y, 1/4 + y)
 
Fourth diagram...

 A = (0, 0)
 B = (1, 0)
 C = (1, 1)
 D = (0, 1)
 E = (1/2, 0)
 F = (0, 1/2)
 G = (1/2 - z, 1/2)
 H = (1/4 + x, 1/4)
 I = (3/4 - y - z/2 - 2xy, 1/4 + y)

Final diagram...

 A = (0, 0)
 B = (1, 0)
 C = (1, 1)
 D = (0, 1)
 E = (1/2, 0)
 F = (0, 1/2 - t)
 G = (1/2 - z, 1/2)
 H = (1/4 + x, 1/4)
 I = (3/4 - y - z/2 - 2xy, 1/4 + y)

Triangles are

 AEH
 EBI
 EIH
 AHF
 HGF
 HIG
 BCG
 FGD
 GCD