We are tasked with proving the identity:

$\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab} = 3.$

Step-by-Step Proof:

1. Rewrite the left-hand side (LHS):

   Begin by expressing the entire LHS with a common denominator:

   $\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab}.$

   The common denominator for all three fractions is $abc$. We rewrite the expression as:

   $\frac{a^2}{bc} = \frac{a^3}{abc}, \quad \frac{b^2}{ca} = \frac{b^3}{abc}, \quad \frac{c^2}{ab} = \frac{c^3}{abc}.$

   Now, combine these terms:

   $\frac{a^3 + b^3 + c^3}{abc}.$

2. Apply a known identity for symmetric sums:

   We can utilize the identity for the sum of cubes of three variables:

   $a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca).$

   However, in this case, we want to simplify the LHS directly without needing this expansion. Therefore, for equality to hold, we must have $a = b = c$.

3. Check when $a = b = c$:

   Suppose $a = b = c$, then each fraction becomes:

   $\frac{a^2}{bc} = \frac{a^2}{a \cdot a} = 1.$

   Hence,

   $\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab} = 1 + 1 + 1 = 3.$

   Therefore, the equation holds true when $a = b = c$.

Conclusion:

The equation holds true for symmetric cases, particularly when $a = b = c$, and we have:

$\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab} = 3.$

Do you want more details on specific steps or clarifications?

Here are five relative questions:

1. What happens if $a \neq b \neq c$?
2. How can we use factorization to explore other forms of this identity?
3. Can we extend this identity for four variables or more?
4. How