In the following, I try to describe the way for deriving the wanted formula (geometric mean value).
Start with the classical 2nd-order bandpass function (involving the parameter pole quality factor Qp and pole frequency wp).
Replace the variable w by wc (3dB cut-off)and - at the same time - set the magnitude of the transfer function to A=Amax/SQRT(2).
As a result, you have a quadratic equation for wc which can be solved.
The result is an equation for the two cut-off frequencies of the form:
wc1=-X+SQRT(X²+wp²) and wc2=+X+SQRT(X2+wp²) with X=wp/2Qp.
EDIT 1: I forgot to mention that the maximum gain Amax at center frequency is the ratio of the numerator and the midterm (jw) of the denominator (because the most left and the most right terms cancel each other at w=wp).
EDIT 2: Here comes a more desriptive explanation of the fact that the distance of both cut-off frequencies (wc1, wc2) to the center frequency wo is different:
The bandpass transfer function is zero for (a) infinite frequencies as well as (b) for w=0. It is clear that the "way" from the center frequncy wo to infinite frequencies is much larger (infinite) if comnpared with the distance to w=0.
That means: The magnitude decrease in the direction to larger frequencies is "smoother" than in the direction to w=0. For this reason, the frequency difference between upper cut-off frequency wc2 and the center frequency wo (wc2-wo) is larger than the difference (wo-wc1).
For this reason (no symmetry to both sides of wo) the center frequency wo is NOT the arithmetic mean value of both cut-off frequcies. (The transfer function magnitude plot vs. frequency looks symmetrical only in case of a logarithmic frequency scale):