Inscribed and circumscribed quadrilateral

[Andrei1]

I would like to discuss the following problem.

The quadrilateral $$ABCD$$ is inscribed into a circle of given radius $$R.$$ And it is circumscribed to a circle. The tangent points from the second circle produce another quadrilateral $$KLMN$$ such that $$S_{ABCD}=3S_{KLMN}.$$ Also $$\gamma$$ is the angle between diagonals $$AC$$ and $$BD.$$ Find the area of $$ABCD.$$

I have no ideas. I wonder if I have to search any regularities of $$ABCD.$$ All given elements seem to me "distanced" from each other.

 

[earboth]

I would like to discuss the following problem.

The quadrilateral $$ABCD$$ is inscribed into a circle of given radius $$R.$$ And it is circumscribed to a circle. The tangent points from the second circle produce another quadrilateral $$KLMN$$ such that $$S_{ABCD}=3S_{KLMN}.$$ Also $$\gamma$$ is the angle between diagonals $$AC$$ and $$BD.$$ Find the area of $$ABCD.$$

I have no ideas. I wonder if I have to search any regularities of $$ABCD.$$ All given elements seem to me "distanced" from each other.

I can't give you a complete solution, sorry, but...

1. For symmetry reasons I assumed that the quadrilateral in question must be a trapezium.

2. The diagonals of ABCD and KLMN intersect in the same point.

3. Since I don't know what $$S_{ABCD}$$ means I can't give you any calculations.

 

[Andrei1]

1. For symmetry reasons I assumed that the quadrilateral in question must be a trapezium.
...
3. ... I don't know what $$S_{ABCD}$$ means ...

The red quadrilateral in your picture can also be circumscribed. $$S$$ is the area.

 

[Opalg]

I would like to discuss the following problem.

The quadrilateral $$ABCD$$ is inscribed into a circle of given radius $$R.$$ And it is circumscribed to a circle. The tangent points from the second circle produce another quadrilateral $$KLMN$$ such that $$S_{ABCD}=3S_{KLMN}.$$ Also $$\gamma$$ is the angle between diagonals $$AC$$ and $$BD.$$ Find the area of $$ABCD.$$

I have no ideas. I wonder if I have to search any regularities of $$ABCD.$$ All given elements seem to me "distanced" from each other.

A quadrilateral of this kind is called bicentric. You might find some useful information at Bicentric quadrilateral - Wikipedia, the free encyclopedia.