The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In other words, if $a$ and $b$ are the lengths of the two shorter sides of a right triangle, and $c$ is the length of the hypotenuse, then $a^2 + b^2 = c^2$.
In the case of a triangle with sides of length $A = 2$, $B = 2$, and $C = 2$, we have $A^2 + B^2 = 2^2 + 2^2 = 8$. However, $C^2 = 2^2 = 4$. Therefore, $A^2 + B^2 \neq C^2$, and the Pythagorean theorem does not hold for this triangle.