Given Information:
We are given three vectors a, b, and c, and we are given the condition |a-c| = |b-c|.

Required:
We need to find the value of b^2 in terms of a and c.

Solution:

Step 1: Understanding the Condition
The condition |a-c| = |b-c| means that the magnitudes of the vectors a-c and b-c are equal. This implies that the vectors a-c and b-c have the same length.

Step 2: Expanding the Magnitudes
We can expand the magnitudes and rewrite the given condition as:
√((a-c) • (a-c)) = √((b-c) • (b-c))

Step 3: Simplifying the Equation
Squaring both sides of the equation, we get:
(a-c) • (a-c) = (b-c) • (b-c)

Expanding the dot products, we have:
a • a - 2(a • c) + c • c = b • b - 2(b • c) + c • c

The terms (a • c) and (c • c) are scalars, so they can be moved to the right side of the equation:
a • a - b • b = 2(b • c) - 2(a • c)

Rearranging the equation, we obtain:
b • b - a • a = 2(a • c - b • c)

Step 4: Finding b^2
Adding a • a to both sides of the equation, we get:
b • b = a • a + 2(a • c - b • c)

Step 5: Simplifying the Equation
We can rewrite the equation in terms of vector operations:
b • b = |a|^2 + 2(a • c - b • c)

Step 6: Rearranging the Equation
Subtracting 2(a • c) from both sides of the equation, we have:
b • b - 2(a • c) = |a|^2 - 2(b • c)

Step 7: Expanding the Dot Product
Expanding the dot products, we get:
b • (b - 2a) = |a|^2 - 2(b • c)

Step 8: Finding b^2
Dividing both sides of the equation by (b - 2a), we obtain:
b = (|a|^2 - 2(b • c)) / (b - 2a)

Therefore, the value of b^2 is:
b^2 = (|a|^2 - 2(b • c)) / (b - 2a)