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### Ellipses touching a circle [closed]

This is a crosspost from math.stackexchange, where the question did not find an answer.
Given a circle and two points $A$, $B$ in the plane, how do I find an ellipse with focal points $A$ and $B$ ...

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### Elementary problem about triangles inside a convex polygon

Let P be a convex polygon with area A(P), and to each side of P, attach the largest area triangle possible that lies entirely within P. Must the sum S(P) of the areas of these triangles always satisfy ...

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### Is there a simple criterion to determine if two parallelograms intersect?

see title.
assume we are given two parallelograms in the plane. how can I check if the intersection is nonempty?
note that I do not need to actually find the intersection.

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### Triangle area on surfaces of constant curvature

I am looking for an elementary derivation of the formula for the area of a geodesic triangle lying in a surface of constant curvature $\kappa$, depending on the angles and side length.
Of course, the ...

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### Definition of area

I am looking for an attractive, but rigorous definition of area;
say in Euclidean plane. Probably there is no short definition. It is OK to make it even longer, but can it be built from useful parts ...