Formula to extract correlation

Charles Angelson writes:

So say you have some variable A, which correlates with B at x. Say you find a variable C where the correlation between C & B is equal to the correlation between C & A. How much will variable C correlate with variable B?
Let the pearson correlation between C & B be y, and the pearson correlation between A & B be x.
y = [1 + x] / sqrt(2x + 2)

EDIT:
I guess I should've specified. The purpose of this is for *generating* C from A & B where A & B explain 100% of variance in C.

I have an update to this.

Given:

-correlation of ‘X’ between A and B

-A and B explain ‘R’ of the variance in C

-A and B equally correlate with C

The formula to calculate the correlation between A|B and C is:

(1 + X)/sqrt(2X + 2)*sqrt(R)

proof: run this code

normult <- c(0, 0.1, 0.25, 0.5, 0.75, 1, 1.5, 2, 2.5, 3, 3.5, 4, 5, 6, 7, 8, 9, 10)
bmult <- c(0, 0.1, 0.25, 0.5, 0.75, 1, 1.5, 2, 2.5, 3, 3.5, 4, 5, 6, 7, 8, 9, 10)

nulls <- rep(0, 324)

dftries <- data.frame(nulls)
dftries$corab <- NA
dftries$corcba <- NA
dftries$r2 <- NA

i = 1
for(nm in normult) {
  for(bm in bmult) {
    A = rnorm(700000)
    B = rnorm(700000) + A*bm
    C = rnorm(700000)*nm + A + B
    ls  <- lm(C ~ A + B)
    lsum <- summary(ls)
    
    dftries$corab[i] <- cor(A, B)
    dftries$corcba[i] <- (cor(C, B) + cor(C, A))/2
    dftries$r2[i] <- lsum$r.squared
    
    i = i+1
  }
}

dftries$y = (1 + dftries$corab)/sqrt(2*dftries$corab + 2)*sqrt(dftries$r2)
ls <- lm(data=dftries, corcba ~ y)
summary(ls)