To calculate the correlation between the residuals of A|B (the variance in A not explained by B) and C, you can use the partial correlation formula. In this case, since A and B equally correlate with C and explain "R" of the variance in C, you can use the following formula:

Partial correlation (A|B and C) = (ρ_AC - ρ_AB * ρ_BC) / √[(1 - ρ_AB^2) * (1 - ρ_BC^2)]

where:

ρ_AC is the correlation between A and C,

ρ_AB is the correlation between A and B (given as "X"),

ρ_BC is the correlation between B and C.

Given that A and B equally correlate with C, we can assume ρ_AC = ρ_BC. Therefore, the formula simplifies to:

Partial correlation (A|B and C) = (ρ_AC - ρ_AB * ρ_AC) / √[(1 - ρ_AB^2) * (1 - ρ_AC^2)]

Here is the further simplified formula:

Partial correlation (A|B and C) = (1 - X) * √(R / (2 - 2X^2))

To calculate the correlation between the residuals of A|B (the variance in A not explained by B) and C, you can use the partial correlation formula. In this case, since A and B equally correlate with C and explain "R" of the variance in C, you can use the following formula:

Partial correlation (A|B and C) = (ρ_AC - ρ_AB * ρ_BC) / √[(1 - ρ_AB^2) * (1 - ρ_BC^2)]

where:

ρ_AC is the correlation between A and C,

ρ_AB is the correlation between A and B (given as "X"),

ρ_BC is the correlation between B and C.

Given that A and B equally correlate with C, we can assume ρ_AC = ρ_BC. Therefore, the formula simplifies to:

Partial correlation (A|B and C) = (ρ_AC - ρ_AB * ρ_AC) / √[(1 - ρ_AB^2) * (1 - ρ_AC^2)]

Here is the further simplified formula:

Partial correlation (A|B and C) = (1 - X) * √(R / (2 - 2X^2))

Proof: It Just Works^tm.