Nesting some triangles

Geometry Level 3

Given that right ΔA0B0C0\Delta A_0B_0C_0 has legs a0a_0 and b0b_0 such that a0=b0=2a_0=b_0=2. Another right triangle ΔA1B0C1\Delta A_1B_0C_1 is drawn from the hypotenuse of ΔA0B0C0\Delta A_0B_0C_0 such that its non-adjacent perpendicular side is drawn 23c0\frac{2}{3} c_0 from B0B_0. Then, another right triangle ΔA2B0C2\Delta A_2B_0C_2 is drawn from the hypotenuse of ΔA1B0C1\Delta A_1B_0C_1 such that its non-adjacent perpendicular side is drawn 23c1\frac{2}{3} c_1 from B0B_0 and so on, as is depicted by the image above.

If all the triangles drawn are similar to one another, what is the sum of areas of all these triangles drawn up to infinity?

Clarification: Uppercase letters denote vertices; lowercase letters denote side lengths.

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