Given that $a$, $b$, and $c$ are in continued proportion, this implies the following relationship:

$\frac{a}{b} = \frac{b}{c} \quad \text{or equivalently} \quad b^2 = ac.$

We need to prove that:

$(a^2 + b^2)(b^2 + c^2) = (ab + bc)^2.$

Step 1: Express everything in terms of $a$ and $c$

From the given proportion, we know that $b^2 = ac$. Let's now substitute $b^2 = ac$ into both sides of the equation and simplify.

Step 2: Expand the Left-Hand Side (LHS)

We expand $(a^2 + b^2)(b^2 + c^2)$:

$(a^2 + b^2)(b^2 + c^2) = a^2b^2 + a^2c^2 + b^4 + b^2c^2.$

Substituting $b^2 = ac$ into this expression:

$= a^2(ac) + a^2c^2 + (ac)^2 + (ac)c^2.$

Simplifying:

$= a^3c + a^2c^2 + a^2c^2 + a c^3.$

Combine like terms:

$= a^3c + 2a^2c^2 + ac^3.$

Step 3: Expand the Right-Hand Side (RHS)

Now let's expand the RHS, $(ab + bc)^2$:

$(ab + bc)^2 = (ab)^2 + 2(ab)(bc) + (bc)^2.$

Simplifying:

$= a^2b^2 + 2ab \cdot bc + b^2c^2.$

Again, using $b^2 = ac$, we substitute:

$= a^2(ac) + 2ab \cdot bc + (ac)c^2.$

Simplifying:

$= a^3c + 2a b^2 c + a c^3.$

Since $b^2 = ac$, this becomes:

$= a^3c + 2a (ac)c + ac^3.$

Simplifying:

$= a^3c + 2a^2c^2 + ac^3.$

Step 4: Conclusion

We see that both sides of the equation are equal:

$a^3c + 2a^2c^2 + ac^3 = a^3c + 2a^2c^2 + ac^3.$

Thus, we have proved that:

$(a^2 + b^2)(b^2 + c^2) = (ab + bc)^2.$

Do you want more details or have any questions? Here are some additional questions to explore:

1. What are continued proportions, and how are they used in geometry?
2. How does the relationship $b^2 = ac$ affect the proof process?
3. Can we generalize this result to more than three variables in continued proportion?
4. How does this identity relate to real-world applications in physics or engineering?
5. What happens if $a$, $b$, and $c$ are negative or complex numbers?

Tip: When solving algebraic identities, expanding both sides of the equation can help reveal whether they are equal.