TITLE: Area of an acute triangle, given two sides and an altitude
QUESTION [5 upvotes]: $12$, $13$, $15$ are the lengths (perhaps not in order) of two sides of an acute-angled triangle and of the height over the third side of triangle. Find the area of the triangle (no calculator allowed).

REPLY [6 votes]: Not an answer, but interesting stuff too long for a comment.
The 13-14-15 triangle was famous when I was on the math team in high school in New York in the early fifties. It has an integral altitude (12) to side 14.
To see why, build it by gluing together the length 12 sides of the 9-12-15 and 5-12-13 right triangles.
This construction generalizes in the obvious way.
High school competitions then were a lot easier than they are now.