Analysis:
Define the points A(0,0), B(a,0), C(a,b), and D(0,b).
Let P(a,y) and Q(x,b). If s is the length of a side
of equilateral triangle APQ, then
s^2 = a^2 + y^2 = x^2 + b^2 = (a - x)^2 + (b - y)^2
Eliminating y from these equations gives a quartic in x
(x^2 + b^2)*(x^2 - 4*a*x + 4*a^2 - 3*b^2) = 0
having two real roots,
x = 2*a - b*sqrt(3)
==> s^2 = 4*(a^2 + b^2 - a*b*sqrt(3))
==> Area(AEF) = sqrt(3)*(a^2 + b^2 - a*b*sqrt(3))
and
x = 2*a + b*sqrt(3)
==> s^2 = 4*(a^2 + b^2 + a*b*sqrt(3))
==> Area(AGH) = sqrt(3)*(a^2 + b^2 + a*b*sqrt(3))
Therefore,
Area(AEF) * Area(AGH) = 3*[a^4 - (a*b)^2 + b^4]
Construction of the triangles:
Construct circles of radii b and centers B and C intersecting
at two points. Construct rays from point A through these two
intersections and intersecting line CD at points F and H.
Construct circles of radii a and centers C and D intersecting
at two points. Construct rays from point A through these two
intersections and intersecting line BC at points E and G.
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