There are lots of things to investigate. We hope you will find
something interesting to contribute. Over the next few months we'll
publish different bits and pieces and different methods until it
adds up to a 'big picture' of this problem.
Squares are constructed on the sides of a triangle and the
outer vertices are joined as shown in the diagram.
This is a
problem that you might like to investigate using dynamic geometry
software but whatever conjectures you are led to by the software
will still need to be proved.
(1) Prove that all four triangles have the same area. (For
convenience in referring to the diagram label the triangles $D, r,
s$ and $t$ and angles $A, B, C, x, y$ and $z$ as shown).

(2) Construct squares on the outer edges of the triangles and
join the outer vertices of the squares to form three
quadrilaterals. Find the angles in these quadrilaterals.
Try tessellating these quadrilaterals with copies of the four
triangles, fitting triangles into the quadrilaterals like pieces of
a jigsaw. What can you prove about these quadrilaterals and their
areas?
(3) Construct another band of squares and quadrilaterals in
the same way. Find the outer angles in the quadrilaterals in this
band.

(4) If this construction is repeated indefinitely building
bands of squares and quadrilaterals, ring upon ring spreading
outwards, what else can you discover?
Thank you Geoff Faux for
introducing this problem to us at the ATM teachers' conference last
April where there was a lot of interest in it.
