Correlation reverses when deflated by the same amount Assume that we have two variables, $a_{i}$ and $b_{i},$measured
across individuals $i.$ They have a strong, positive correlation,
say $corr(a_{i},b_{i})\approx0.9$.  Now, there is a another variable,
$c_{i}$ , which varies across individuals.  Note also that
$$
0<a_{i}\leq b_{i}\leq c_{i}
$$
We compute, then, for
each individual:
$$
A_{i}=\frac{a_{i}}{c_{i}}
$$
and
$$
B_{i}=\frac{b_{i}}{c_{i}}
$$
We notice that $corr(A_{i},B_{i})$ becomes strongly negative! This
is a bit perplexing, but the only reasonable explanation I can come
up with is this- across individuals, when $b_{i}$ increases, although
$a_{i}$ increases, $c_{i}$ increases by more than $a_{i}$ increases
on average, rendering this correlation negative. Is this reasoning
correct? Is there perhaps a formal condition linking $a,b$ and $c$
which can capture the essence of this reverse in correlation?

This can happen when the $c_i$ are positively correlated with $a_i+b_i.$  The geometry of the $(a_i,b_i)$ scatterplot reveals that such behavior should not be surprising.

Without any loss of generality we may scale all the $a_i$ by some positive value, all the $b_i$ by some other positive value, and all the $c_i$ by yet a third positive value, without changing any of the conditions.
Thus, it's no restriction to imagine the $(a_i,b_i)$ scatterplot takes on a classic "cigar" shape, with its axis oriented around 45 degrees.
The rescaling by the $c_i$ moves each point to another one on the line connecting it to the origin.  Thus, we need to envision a situation in which the points in the original cigar can be moved in this way to fall along a cigar-shaped cloud that is oriented negatively.  This is easily done: pick any nonzero number $\mu$ and for each point $i$ set

$$c_i \approx (a_i + b_i)/\mu.$$

The projected point
$$\left(\frac{a_i}{c_i},\frac{b_i}{c_i}\right) = \mu\left(\frac{a_i}{a_i+b_i},\frac{b_i}{a_i+b_i}\right)$$
will lie close to the line $a_i+b_i=\mu,$ which has negative slope.  Thus, you can achieve any negative correlation, even $-1.0$ exactly, by varying the closeness of the approximation.  The value $\mu\sqrt{2}$ will be the typical distance between the new points and the line $b = -a.$
Here is an example to illustrate.  The $n=30$ original points with correlation coefficient $0.9$ are plotted as open circles.  Arrows project them to new solid red points with correlation coefficient $-0.8.$  Those points all lie near the line $a+b \approx 5.57$ (the median of all the $a$ and $b$ coordinates).

