Opposite Earth
The North American and South American continents are offset so that the east coast of North America more or less lines up with the west coast of South America. This being the case, one might think of the "center" of the Western Hemisphere as being the location where the equator intersects this line, that is, the point along the western coast of South America where the equator crosses. This would be a point approximately halfway between Portoviejo and Esmereldas along the northern coast of Ecuador. Where would the point on the earth be located that is exactly opposite this point (i.e., the "middle" of the Eastern Hemisphere)?

Answer: On the island of Sumatra in Indonesia.

Explanation: The longitude of that point on the coast of Ecuador is approximately 80^o west longitude. Opposite of this would mean 180^o away, which would be 100^o east longitude. A look at an atlas will reveal that the point along the equator at 100^o east longitude is located on Sumatra, Indonesia (about 80 miles north of the city of Padang).