We are given that $a$, $b$, and $c$ are the sides of a triangle, and the equation

$a^2 + b^2 + c^2 = a^2 b^2 c^2$

is given as a condition. We are asked to prove that the maximum area of this triangle is $\frac{3}{4}$, and that this maximum is attained only when $a = b = c$.

Step 1: Analyze the Condition $a^2 + b^2 + c^2 = a^2b^2c^2$

First, rewrite the given equation in a more tractable form. Let’s denote the sides of the triangle as $a$, $b$, and $c$. From the equation:

$a^2 + b^2 + c^2 = a^2b^2c^2$

This is a non-standard condition, but we aim to explore this relation with known properties of triangles.

Step 2: Consider the Symmetry of the Problem

Notice that the equation is symmetric with respect to $a$, $b$, and $c$. This suggests that a situation where $a = b = c$ might give us important insight into the behavior of the equation.

If we assume that $a = b = c = x$, then the equation becomes:

$3x^2 = x^6$

Simplifying this, we get:

$x^6 - 3x^2 = 0$

Factoring out $x^2$, we obtain:

$x^2(x^4 - 3) = 0$

Thus, either $x^2 = 0$ (which is not possible since $a$, $b$, and $c$ are sides of a triangle and must be positive), or:

$x^4 = 3$

Taking the square root of both sides:

$x^2 = \sqrt{3}, \quad x = \sqrt[4]{3}$

Thus, $a = b = c = \sqrt[4]{3}$ is a potential solution when the sides are equal.

Step 3: Area of the Triangle when $a = b = c$

For an equilateral triangle, the area $A$ is given by:

$A = \frac{\sqrt{3}}{4} a^2$

Substituting $a = \sqrt[4]{3}$:

$A = \frac{\sqrt{3}}{4} \left( \sqrt[4]{3} \right)^2 = \frac{\sqrt{3}}{4} \times \sqrt{3} = \frac{3}{4}$

Thus, the area of the triangle when $a = b = c = \sqrt[4]{3}$ is $\frac{3}{4}$.

Step 4: Uniqueness of the Maximum Area

From the symmetry of the given equation $a^2 + b^2 + c^2 = a^2b^2c^2$, and the fact that it leads to a specific solution for equal sides, we conclude that the maximum area is achieved only when $a = b = c = \sqrt[4]{3}$.

Therefore, the maximum area of the triangle is indeed $\frac{3}{4}$, and equality holds only when $a = b = c$.

Conclusion:

The maximum area of the triangle is $\frac{3}{4}$, and this maximum is attained only when $a = b = c$.

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Would you like more details on any part of the proof, or further clarifications? Here are five related questions you could explore:

1. What is the geometric significance of the equation $a^2 + b^2 + c^2 = a^2b^2c^2$?
2. How do you compute the area of a triangle for arbitrary side lengths?
3. Can this result be generalized to other types of triangles besides equilateral ones?
4. What other properties are true for a triangle when $a = b = c$?
5. How would you approach the problem if the sides were not equal?

Tip: When proving the maximum or minimum value of a geometric quantity, symmetry often simplifies the analysis.