Triangles ABD and ABC stand on the same base AB, and the vertices C and D lie on a single line drawn parallel to AB. A student claims that area(ABD) : area(ABC) must be the square of AD : AC.
What is the correct value of area(ABD) : area(ABC)?

The two triangles share the base AB, and because C and D lie on a line parallel to AB they also have the same height. Equal base and equal height give equal areas, so the ratio is 1 : 1. The theorem the student quoted (area ratio = square of the side ratio) applies only to SIMILAR triangles, and these two need not be similar at all.