Prediction: Denver Shadows, Sept 22, 2016 Equinox
For 39° 44' 21.119" N / 104° 59' 25.08" W in Denver Colorado on Sept 22, 2016 (the autumnal equinox for 2016 at which point the sun will be directly over the equator) at noon local time (19:00Z) the shadow cast by a plumb 5 meter tall pole should be ~4.12858 meters long (not that we can measure that accurately but should be close). This calculation assumes a Spherical Earth with the sun 93 million miles away.
This location is ~4401481 meters from the equator (found using Google Earth Pro which uses the actual Geo data rather than assuming a simple sphere) which means that, on a Flat Earth, with the Sun 4000 mile away, you would have a shadow length of just 3.4186972 meters.
For that to match a Flat Earth model the sun would have to be just 5330.51 km (or just 3312 miles) high.
But then that measurement cannot match the shadow lengths at other latitudes.
Every latitude will get a different value for the 'height' of the sun to match the shadow length.
Children around the world will all be getting shadow length data on this date as part of the Noon Day Project. You can use their data to show that the shadow lengths simply do match a Flat Earth model.
What exact latitude & longitude are you at? What shadow length predictions do you get?