The acute triangle ABC has sides a>b>c, where A is opposite a and likewise for B and C.
A square is inscribed with vertices on the edges of the triangle.
It is given that only three such squares exist, and that one side of each square is concurrent with the side of a triangle.
Suppose the side length of the squares are p,q,r (with any correspondence you want), and the altitudes of the triangle are a',b',c', which connect the vertices A,B,C with the sides a,b,c respectively.
i) Find expressions for each of the three squares in terms of all the variables described above.
ii) Determine which of the three squares has the largest area.
iii) Hence, or otherwise, find the smallest possible area of any such triangle ABC which encloses a square of unit area.