If a triangle $$ \Delta ABC $$ is inscribed in a circle (with vertices $$ A $$, $$ B $$, and $$ C $$ on the circumference), the area of $$ \Delta ABC $$ will be maximum when:

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For a triangle inscribed in a circle, the area is maximized when the triangle is equilateral. An equilateral triangle ensures that all three sides are equal, i.e., $$ AB = BC = CA $$. This configuration provides the largest possible area for a triangle within the same circle.

Thus, the area of $$ \Delta ABC $$ is maximum when **$$ AB = BC = CA $$**.

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