By way of illustration, consider the bandpass filter below.

The frequency response function is:
\$\small \frac{V2}{V1}= G(j\omega)=\large\frac{R}{R+j(\omega L-\frac{1}{\omega C})}\$
and the gain is:
\$\small G(\omega)=\large \frac{R}{\sqrt{R^2+(\omega L-\frac{1}{\large \omega C})^2}}\$
The resonant frequency, \$\small \omega_r\$, is defined when \$\small \omega L=\large \frac{1}{\omega C}\$, giving \$\omega_r =\frac{1}{\sqrt{LC}}\$, and the gain at resonance is unity.
We can determine the bandwidth of this filter by calculating the two corner frequencies, say \$\small \omega_l\$ and \$\small \omega_u\$, where the gain is 3dB down from the gain at resonance, or in other words where the gain is \$\frac{1}{\sqrt{2}}\$.
From the gain equation, \$\small \omega_l\$ and \$\small \omega_u\$ must be defined by \$\small (\omega L-\large \frac{1}{\omega C})^2=\small R^2\$, and we can determine \$\small \omega_l\$ and \$\small \omega_u\$ by solving the quadratic equation:
\$\small \omega^2 LC-\omega RC-1=0\$, noting that \$\omega\$ must be a positive value.
Thus: \$\omega_l=\frac{\sqrt{R^2C^2+4LC}-RC}{2LC}\$ and \$\omega_u=\frac{\sqrt{R^2C^2+4LC}+RC}{2LC}\$
Now, the geometric mean, \$\omega_c\$, of \$\omega_l\$ and \$\omega_u\$, is defined when: \$\large \frac{\omega_u}{\omega_c}=\frac{\omega_c}{\omega_l}\$, or \${\omega_c}^2=\omega_l \: \omega_u=\frac{1}{LC}={\omega_r}^2\$. Hence $$\large \omega_c=\omega_r$$
That is, the geometric mean of the corner frequencies is the resonant frequency.