TITLE: Partitioning a Rectangle into Congruent Isosceles Triangles
QUESTION [15 upvotes]: Is it possible to partition any rectangle into congruent isosceles triangles?

REPLY [24 votes]: No. Note that the acute angle of your triangle must divide $\pi/2$ (look at a corner), so there are countably many such triangles (up to similarity), and hence you get only a countable set of possible ratios of sides.