To determine the relationship between \(a\), \(b\), and \(c\) when the roots of the given equation are equal, let's examine the equation:

The equation provided is:

\( abx^2 + bcx + ca = cax^2 + abx + bc \)

First, let's simplify this equation to find a standard quadratic form. Rearrange the terms to get:

\( abx^2 + bcx + ca - cax^2 - abx - bc = 0 \)

Collecting like terms, it becomes:

\( (ab - ca)x^2 + (bc - ab)x + (ca - bc) = 0 \)

For the roots of this quadratic equation to be equal, the discriminant of this equation must be zero. The discriminant of a quadratic equation \(Ax^2 + Bx + C = 0\) is given by:

\(\Delta = B^2 - 4AC\)

Here, \(A = (ab - ca)\), \(B = (bc - ab)\), and \(C = (ca - bc)\).

So, the discriminant \(\Delta\) is:

\(\Delta = (bc - ab)^2 - 4(ab - ca)(ca - bc)\)

Setting \(\Delta = 0\) for the roots to be equal, we have:

\((bc - ab)^2 = 4(ab - ca)(ca - bc)\)

To solve this, we recognize that if the equation must hold true for the roots to be equal in all respects, \(a\), \(b\), and \(c\) must satisfy a specific relationship.

Consider the case where \(a\), \(b\), and \(c\) are in Harmonic Progression (HP).

We know numbers \(a\), \(b\), and \(c\) are in HP if the reciprocals \(1/a\), \(1/b\), and \(1/c\) are in Arithmetic Progression (AP). This gives the condition:

\(\frac{2}{b} = \frac{1}{a} + \frac{1}{c}\)

Cross-multiply to verify:

\(2ac = ab + bc\)

This equation matches the symmetry required for the discriminant to be zero in this context, confirming that \(a\), \(b\), and \(c\) are in HP.

Thus, when the roots are equal, \(a\), \(b\), and \(c\) are in Harmonic Progression (HP).