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Given a triangle of sides $a,b,c$ three non concentric and not intersecting circles are to be inscribed in that triangle such that the sum of areas of enclosed circles is maximum..This is an extreme value problem but I am not sure what to begin with...

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The problem is known as Malfatti's marble problem (as opposed to Malfatti's circle problem, which asks for the construction of Malfatti circles). Quoting Wikipedia:

Lob and Richmond (1930), who went back to the original Italian text, observed that for some triangles a larger area can be achieved by a greedy algorithm that inscribes a single circle of maximal radius within the triangle, inscribes a second circle within the largest of the three remaining corners of the triangle, and inscribes a third circle within the largest of the five remaining pieces.

And a bit later on:

Zalgaller and Los' (1994) classified all of the different ways that a set of maximal circles can be packed within a triangle; using their classification, they proved that the greedy algorithm always finds three area-maximizing circles, and they provided a formula for determining which packing is optimal for a given triangle.

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