The title "Triangle Paradox" gives a hint: They are not triangles with or without the hole!
In order for red and green triangles to line up perfectly, they must be similar triangles having the same slope. Yet they aren't.
The triangle makes use of the properties of the Fibonacci numbers. Notice that the triangle is of has sides 13 and 5 (Fibonacci). In particular it is possible to show that (Fn)^2 (the nth Fibonacci number squared) differs from F(n-1) x F(n+1) by 1 always. Sometimes + 1 sometimes - 1. The triangle you show just plays on this. Have a look at a 21 x 21 square and a 34 x 13 rectangle.