Draw any two squares which meet at a common vertex C and join the
adjacent vertices to make two triangles CAB and CDE.
Construct the perpendicular from C to AB, (the altitude of the
triangle). When you extend this line where does it cut DE?
Now bisect the line AB to find the midpoint of this line M. Draw
the median MC of triangle ABC and extend it to cut DE. What do
you notice about the lines MC and DE?
Will you get the same results about the two triangles formed if you
draw squares of different sizes or at different angles to each
other? Make a conjecture about the altitude of one of these
triangles and prove your conjecture.
Thank you Geoff Faux for suggesting this problem.