A dip circle is positioned at the geomagnetic equator. Its needle...

At the geomagnetic equator, the angle of dip is zero, which means the vertical component of the Earth's magnetic field is zero ($B_V = 0$).

The horizontal component of the Earth's magnetic field ($B_H$) acts along the magnetic meridian.

The dip circle is placed such that its needle rotates in a vertical plane perpendicular to the magnetic meridian. The angle between this plane and the magnetic meridian is $\alpha = 90^{{}^{{}^{\circ}}}$.

The horizontal component of the magnetic field in this plane is $B_H' = B_H \cos 90^{{}^{{}^{\circ}}} = 0$.

The vertical component in this plane is $B_V' = B_V = 0$.

Since both the horizontal and vertical components of the magnetic field in the plane of rotation are zero, the effective magnetic field in this plane is zero.

Consequently, the magnetic needle experiences no restoring torque about its axis of rotation. Being balanced at its center of gravity, it will remain in neutral equilibrium and stay in whatever direction it is released.