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Let $\Delta := (A,B,C)$ be a triangle that is defined by three points in the Euclidean plane that are not collinear. Let further $E_{(A,B),\,C},\,E_{(C,A),\,B},\,E_{(B,A),\,B}$ be the set of ellipses with two of the corners of $\Delta$ as their foci and the third on their boundary.

Question:
What is the area of the intersection of $E_{(A,B),\,C},\,E_{(C,A),\,B},\,E_{(B,A),\,B}$ expressed as a function of the triangle's sidelengths $a,\,b,\,c$?

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    $\begingroup$ You might look at Finding ellipse-ellipse intersections in $\mathbb{R}^2$. Calculating the intersection of two ellipses is already a complex calculation, involving finding roots of quartics. $\endgroup$
    – Joseph O'Rourke
    Commented May 8, 2020 at 19:52
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    $\begingroup$ Suppose the points are at $(0,1)$, $(-2,0)$ and $(0,3)$. Then two of the ellipses have intersections at $$x= \frac{\mp 9 \sqrt{26} b+797702 \sqrt{130}-2638959 \sqrt{13}-6690567 \sqrt{10}-29138642}{50442431}$$ $$y= -\frac{3 \left(\pm(-6 \sqrt{65}-12 \sqrt{26} +26 \sqrt{5}+39 \sqrt{2})b+47051 \sqrt{130}-526905 \sqrt{13}-1464372 \sqrt{10}-22007284\right)}{50442431}$$ where $b^2=4201243714808 + 55949265287 \sqrt{10} + 56446668509 \sqrt{13} + 362189770403 \sqrt{130}$, from which I conclude that any formula would be a mess. $\endgroup$
    – user44143
    Commented May 9, 2020 at 23:40

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