If I understand correctly, in countries to the north of the tropics the sun's noon shadow points north.
What happens south of the tropics? Is it pointing south?
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Is it pointing south?
Yes, the shadow points south when you are south of the tropics (south of the Tropic of Capricorn).
In the tropics (between the Tropic of Cancer and the Tropic of Capricorn), the direction of the Sun's shadow depends on the location and the date.
As the angle between the sun and the earth's axis changes during the year, the apparent path of the sun in the sky changes.
Outside the tropics, this change is insufficient to ever bring the sun directly overhead. Regardless of the date, the sun will be seen to be toward the equator at local noon. (Same as saying the shadow points away from the equator).
Within the tropics, the changing path sometimes brings the sun directly overhead. The position at local noon depends on the specific location and the date.
On the December solstice, the sun will be overhead for locations on the Tropic of Capricorn and will cross to the south at local noon for all locations in the tropics (and those further north). During the June solstice, the situation is reversed for the tropics. All locations south of the Tropic of Cancer would see the sun pass to the north.