geometry

I believe that the first (largest) blue areas of the triangle form 3/4^ths of the complete triangle.

The next blue area forms 3/4^ths of 1/4^th of 1/4^th of the triangle.

The next blue area forms 3/4^ths of 1/4^th of 1/4^th of 1/4^th of 1/4^th of the triangle.

Etc.

 

(3/4)  +  (3/4)(1/4)²  +  (3/4)(1/4)⁴  +  (3/4)(1/4)⁶  +  ...

 

This forms a geometric sequence with  first term  =  3/4  and with  common ratio  =  1/16:

Sum  =  (3/4) / [ 1 - 1/16 ]  =  (3/4) / (15/16)  =  4/5.