Ellipses Inside Triangles

For any triangle $T$ in the plane, it can be shown that there is a unique ellipse with largest area that is completely inside $T$.

For a given $n$, consider triangles $T$ such that:
– the vertices of $T$ have integer coordinates with absolute value $\le n$, and
– the foci1 of the largest-area ellipse inside $T$ are $(\sqrt{13},0)$ and $(-\sqrt{13},0)$.
Let $A(n)$ be the sum of the areas of all such triangles.

For example, if $n = 8$, there are two such triangles. Their vertices are $(-4,-3),(-4,3),(8,0)$ and $(4,3),(4,-3),(-8,0)$, and the area of each triangle is $36$. Thus $A(8) = 36 + 36 = 72$.

It can be verified that $A(10) = 252$, $A(100) = 34632$ and $A(1000) = 3529008$.

Find $A(1\,000\,000\,000)$.

1The foci (plural of focus) of an ellipse are two points $A$ and $B$ such that for every point $P$ on the boundary of the ellipse, $AP + PB$ is constant.

The problem described here is a creative one with relation to geometry and number theory, but it’s not a problem that would commonly be solved in a traditional school lesson. This problem seems transcend advanced high school and even most university-level classes. It is more likely to appear on advanced math contests or in research.

Finding A(1,000,000,000) would be computationally intensive and requires a detailed understanding of advanced mathematics, like the properties of ellipses (especially how they relate to enclosing triangles), coordinate geometry, and number theory.

However, I can give you a general direction of how to, potentially, approach this problem:
1. Try to find a rule or pattern based on the information given. We know the foci of the ellipses, hence, we can get the equation of the ellipse in a triangle.
2. Identify the conditions under which the ellipse will be the biggest while inside a triangle. This would perhaps involve techniques from optimization or could potentially involve inherent properties of ellipses and their surrounding triangles.
3. With rules established, for each applicable triangle, we quantify the “area” and then write an algorithm to run through all such triangles in range (-1,000,000,000, 1,000,000,000).
4. Add up all the areas.

Unfortunately, even the general approach detailed above would require a quite high level of mathematical ability, as well as coding capabilities to run through all the possible triangles within the given range.

Note that it is out of capabilities of a general-purpose language model AI in its current state to provide a more specific or detailed solution to this problem. An appropriate approach to solving this problem would potentially involve the use of a mathematical software package or a programming language with built-in algorithms for this kind of mathematical manipulations.

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