Agates mostly occur as nodules in volcanic rocks or waters containing Silica in solution which is possibly derived from the decomposition of Silicates in the lava. The Siliceous matter is deposited in regular layers resulting in the banded or striped formation. It is very unlikely that you would ever find an exact duplicate of any type of Agate, Brazilian or other.
Goethite is an Iron bearing Oxide mineral found in soil and other low temperature environments. Historically, Goethite has been used as a paint pigment found on cave paintings in France. Its main modern use is that of an Iron Ore content or objects using Iron Ore. Goethite is found worldwide usually in the form of concretions, stalactitic formations, oolites, reniform or botryoidal accumulations.