if a , b , c are in continued proportion, then prove that (a²+b²) (b²+c²) = (ab+bc)^2

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.

If a, b, c are in continued proportion, then prove that (a² + b²)(b² + c²) = (ab + bc)².

Explore the detailed solution to the algebraic identity (a² + b²)(b² + c²) = (ab + bc)², given that a, b, and c are in continued proportion. This proof uses key algebraic concepts, including polynomial expansion and properties of continued proportions, suitable for students in Grades 10-12.

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