  We investigate the possibility of determining whether microlensing objects
towards the Large Magellanic Cloud (LMC) are in a Galactic thick disc, or are
in a Galactic halo, by using parallax measurements with an Earth-radius scale
baseline. Our method makes use of EAGLE (Extremely Amplified Gravitational
LEnsing) events which are microlensing events with an invisible faint source.
We show that the rate of EAGLE events is as high as that of normal microlensing
events, even if they are caused by dark stars in the Galactic thick disc. We
explore the possibility of measuring the parallax effect in EAGLE events
towards the LMC by using the {\it Hubble Space Telescope} (HST) or the {\it
Very Large Telescope} (VLT). We find that EAGLE events enlarge the opportunity
of parallax measurements by $4 \sim 10$ times relative to that in normal
microlensing events. We show that the parallax effect can be measured in
$\sim75%$ (from the HST) and $\sim 60%$ (from the VLT) of all EAGLE events if
most lenses are stars in the Galactic thick or thin disc, while $\sim 20%$
(from the HST) and $\sim 10%$ (from the VLT) can be measured if most lenses are
halo MACHOs. In combination with the finite source size effect observations, we
can strongly constrain the location of lenses.
